2. Lie invariance algebra

The classical infinitesimal approach results in the well-known algorithm for computing the maximal Lie-symmetry algebras of systems of differential equations [Reference Bluman, Cheviakov and Anco9, Reference Bluman and Kumei10, Reference Olver25]. The maximal Lie invariance algebra of the equation (2) is (see, e.g. [Reference Kovalenko, Kopas and Stogniy20])

\begin{align*} \mathfrak{g}\,:\!=\langle \mathcal{P}^t,\,\mathcal{D},\,\mathcal{K},\, \mathcal{P}^3,\,\mathcal{P}^2,\,\mathcal{P}^1,\,\mathcal{P}^0,\,\mathcal{I},\mathcal{Z}(\,f)\rangle, \end{align*}

where

\begin{align*} &\mathcal{P}^t =\partial _t,\ \ \mathcal{D} =2t\partial _t+x\partial _x+3y\partial _y-2u\partial _u,\ \ \mathcal{K} =t^2\partial _t+(tx+3y)\partial _x+3ty\partial _y-(x^2\!+2t)u\partial _u,\\ &\mathcal{P}^3 =3t^2\partial _x+t^3\partial _y+3(y-tx)u\partial _u,\ \ \mathcal{P}^2 =2t\partial _x+t^2\partial _y-xu\partial _u,\ \ \mathcal{P}^1 =\partial _x+t\partial _y,\ \ \mathcal{P}^0 =\partial _y,\\ &\mathcal{I} =u\partial _u,\quad \mathcal{Z}(\,f)=\,f(t,x,y)\partial _u. \end{align*}

Here the parameter function $\,f$ of $(t,x,y)$ runs through the solution set of the equation (2).

The vector fields $\mathcal{Z}(\,f)$ constitute the infinite-dimensional abelian ideal $\mathfrak{g}^{\textrm{lin}}$ of $\mathfrak{g}$ associated with the linear superposition of solutions of (2), $\mathfrak{g}^{\textrm{lin}}\,:\!=\{\mathcal{Z}(\,f)\}$ . Thus, the algebra $\mathfrak{g}$ can be represented as a semidirect sum, $\mathfrak{g}=\mathfrak{g}^{\textrm{ess}}\mathbin{\mbox{${\,\,\rule{.4pt}{1.07ex}}{\!\!\in }$}}\mathfrak{g}^{\textrm{lin}}$ , where

(4) \begin{align} \mathfrak{g}^{\textrm{ess}}=\langle \mathcal{P}^t,\mathcal{D},\mathcal{K},\mathcal{P}^3,\mathcal{P}^2,\mathcal{P}^1,\mathcal{P}^0,\mathcal{I}\rangle \end{align}

is an (eight-dimensional) subalgebra of $\mathfrak{g}$ , called the essential Lie invariance algebra of (2).

Up to the skew-symmetry of the Lie bracket, the nonzero commutation relations between the basis vector fields of $\mathfrak{g}^{\textrm{ess}}$ are the following:

\begin{align*} &[\mathcal{P}^t,\mathcal{D}] = 2\mathcal{P}^t,\quad [\mathcal{P}^t,\mathcal{K}] = \mathcal{D},\quad [\mathcal{D}, \mathcal{K}] = 2\mathcal{K},\\ &[\mathcal{P}^t,\mathcal{P}^3]= 3\mathcal{P}^2,\quad [\mathcal{P}^t,\mathcal{P}^2]= 2\mathcal{P}^1,\quad [\mathcal{P}^t,\mathcal{P}^1]= \mathcal{P}^0,\\ &[\mathcal{D},\mathcal{P}^3] = 3\mathcal{P}^3,\quad [\mathcal{D},\mathcal{P}^2] = \mathcal{P}^2,\quad [\mathcal{D},\mathcal{P}^1] =- \mathcal{P}^1,\quad [\mathcal{D},\mathcal{P}^0] =-3\mathcal{P}^0,\\ &[\mathcal{K},\mathcal{P}^2] =- \mathcal{P}^3,\quad [\mathcal{K},\mathcal{P}^1] =-2\mathcal{P}^2,\quad [\mathcal{K},\mathcal{P}^0] =-3\mathcal{P}^1,\\ &[\mathcal{P}^1,\mathcal{P}^2]=- \mathcal{I},\quad [\mathcal{P}^0,\mathcal{P}^3]= 3\mathcal{I}. \end{align*}

The algebra $\mathfrak{g}^{\textrm{ess}}$ is nonsolvable. Its Levi decomposition is given by $\mathfrak{g}^{\textrm{ess}}=\mathfrak f\mathbin{\mbox{${\,\,\rule{.4pt}{1.07ex}}{\!\!\in }$}} \mathfrak r$ , where the radical $\mathfrak r$ of $\mathfrak{g}^{\textrm{ess}}$ coincides with the nilradical of $\mathfrak{g}^{\textrm{ess}}$ and is spanned by the vector fields $\mathcal{P}^3$ , $\mathcal{P}^2$ , $\mathcal{P}^1$ , $\mathcal{P}^0$ and $\mathcal{I}$ . The Levi factor $\mathfrak f=\langle \mathcal{P}^t,\mathcal{D},\mathcal{K}\rangle$ of $\mathfrak{g}^{\textrm{ess}}$ is isomorphic to $\textrm{sl}(2,\mathbb R)$ , the radical $\mathfrak r$ of $\mathfrak{g}^{\textrm{ess}}$ is isomorphic to the rank-two Heisenberg algebra $\textrm{h}(2,\mathbb R)$ , and the real representation of the Levi factor $\mathfrak f$ on the radical $\mathfrak r$ coincides, in the basis $(\mathcal{P}^3,\mathcal{P}^2,\mathcal{P}^1,\mathcal{P}^0,\mathcal{I})$ , with the real representation $\rho _3\oplus \rho _0$ of $\textrm{sl}(2,\mathbb R)$ . Here $\rho _n$ is the standard real irreducible representation of $\textrm{sl}(2,\mathbb R)$ in the $(n+1)$ -dimensional vector space. More specifically,

\begin{equation*} \rho _n( \mathcal {P}^t)_{ij}=(n-j)\delta _{i,j+1},\quad \rho _n( \mathcal {D})_{ij} =(n-2j)\delta _{ij},\quad \rho _n({-}\mathcal {K})_{ij} =j\delta _{i+1,j}, \end{equation*}

where $i,j\in \{1,2,\dots,n+1\}$ , $n\in \mathbb N\cup \{0\}$ , and $\delta _{kl}$ is the Kronecker delta, that is, $\delta _{kl}=1$ if $k=l$ and $\delta _{kl}=0$ otherwise, $k,l\in \mathbb Z$ . Thus, the entire algebra $\mathfrak{g}^{\textrm{ess}}$ is isomorphic to the algebra $L_{8,19}$ from the classification of indecomposable Lie algebras of dimensions up to eight with nontrivial Levi decompositions, which was carried out in [Reference Turkowski38].

Lie algebras whose Levi factors are isomorphic to the algebra $\textrm{sl}(2,\mathbb R)$ often arise within the field of group analysis of differential equations as Lie invariance algebras of parabolic partial differential equations. At the same time, the action of Levi factors on the corresponding radicals is usually described in terms of the representations $\rho _0$ , $\rho _1$ , $\rho _2$ or their direct sums. To the best of our knowledge, algebras similar to $\mathfrak{g}^{\textrm{ess}}$ were never considered in group analysis from the point of view of their subalgebra structure.

3. Complete point symmetry pseudogroup

We start computing the complete point symmetry pseudogroup $G$ of the equation (2) by presenting the exhaustive description of the equivalence groupoid of the class $\bar{\mathcal{F}}$ . Then we use special properties of this groupoid for deriving an explicit representation for elements of $G$ . See [Reference Bihlo, Dos Santos Cardoso-Bihlo and Popovych4, Reference Bihlo and Popovych6, Reference Opanasenko, Bihlo and Popovych26, Reference Popovych31, Reference Popovych, Kunzinger and Eshraghi33, Reference Vaneeva, Bihlo and Popovych39] and references therein for definitions and theoretical results on various structures constituted by point transformations within classes of differential equations.

Theorem 1. The class $\bar{\mathcal{F}}$ is normalised. Its (usual) equivalence pseudogroup $G^\sim _{\bar{\mathcal{F}}}$ consists of the point transformations with the components

(5a) \begin{align} \qquad \tilde t=T(t,y), \quad \tilde x=X(t,x,y), \quad \tilde y=Y(t,y), \quad \tilde u=U^1(t,x,y)u + U^0(t,x,y), \end{align}

(5b) \begin{align}\tilde A^0=\dfrac{A^0}{T_t+BT_y}-\dfrac{A^1}{T_t+BT_y}\dfrac{U^1_x}{U^1}+\dfrac{A^2}{T_t+BT_y}\bigg (\left (\frac{U^1_x}{U^1}\right )^2- \left (\frac{U^1_x}{U^1}\right )_x\bigg )+\dfrac 1{U^1}\dfrac{U^1_t+BU^1_y}{T_t+BT_y}, \end{align}

(5c) \begin{align} \tilde A^1=A^1\dfrac{X_x}{T_t+BT_y}-\dfrac{X_t+BX_y}{T_t+BT_y}+A^2\dfrac{X_{xx}-2X_xU^1_x/U^1}{T_t+BT_y}, \end{align}

(5d) \begin{align} \tilde A^2=A^2\dfrac{X_x^2}{T_t+BT_y}, \quad \tilde B=\dfrac{Y_t+BY_y}{T_t+BT_y}, \end{align}

(5e) \begin{align} \tilde C=\dfrac{U^1}{T_t+BT_y}\left (C-E\dfrac{U^0}{U^1}\right ), \end{align}

where $T$ , $X$ , $Y$ , $U^1$ and $U^0$ are arbitrary smooth functions of their arguments with $(T_tY_y-T_yY_t)X_xU^1\neq 0$ , and $E\,:\!=\partial _t+B\partial _y-A^2\partial _{xx}-A^1\partial _x-A^0$ .

The proof of this theorem is beyond the subject of the present paper and will be presented elsewhere.

The equation (2) corresponds to the value $(x,1,0,0,0)=\!:\,\bar \theta _0$ of the arbitrary-element tuple $\bar \theta =(B,A^2,A^1,A^0,C)$ of the class $\bar{\mathcal{F}}$ . The vertex group $\mathcal{G}_{\bar \theta _0}\,:\!=\mathcal{G}^\sim _{\bar{\mathcal{F}}}(\bar \theta _0,\bar \theta _0)$ is the set of admissible transformations of the class $\bar{\mathcal{F}}$ with $\bar \theta _0$ as both their source and target, $\mathcal{G}_{\bar \theta _0}=\left\{(\bar \theta _0,\Phi,\bar \theta _0)\mid \Phi \in G\right\}$ . The normalisation of the class $\bar{\mathcal{F}}$ means that its equivalence groupoid coincides with the action groupoid of the pseudogroup $G^\sim _{\bar{\mathcal{F}}}$ , and thus, the latter groupoid necessarily contains the vertex group $\mathcal{G}_{\bar \theta _0}$ . This argument allows us to use Theorem 1 in the course of computing the pseudogroup $G$ .

Theorem 2. The complete point symmetry pseudogroup $G$ of the remarkable Fokker–Planck equation (2) consists of the transformations of the form

(6) \begin{align} \begin{split} &\tilde t=\frac{\alpha t+\beta }{\gamma t+\delta }, \quad \tilde x=\frac{\hat x}{\gamma t+\delta } -\frac{3\gamma \hat y}{(\gamma t+\delta )^2}, \quad \tilde y=\frac{\hat y}{(\gamma t+\delta )^3}, \\[1ex] &\tilde u=\sigma (\gamma t+\delta )^2\exp \left ( \frac{\gamma \hat x^2}{\gamma t+\delta } -\frac{3\gamma ^2\hat x\hat y}{(\gamma t+\delta )^2} +\frac{3\gamma ^3\hat y^2}{(\gamma t+\delta )^3} \right ) \\ &\hphantom{\tilde u={}} \times \exp\!\left ( 3\lambda _3(y-tx)-\lambda _2x-(3\lambda _3^2t^3+3\lambda _3\lambda _2t^2+\lambda _2^2t) \right ) \left (u+\,f(t,x,y)\right ), \end{split} \end{align}

where $\hat x\,:\!=x+3\lambda _3t^2+2\lambda _2t+\lambda _1$ , $\hat y\,:\!=y+\lambda _3t^3+\lambda _2t^2+\lambda _1t+\lambda _0$ ; $\alpha$ , $\beta$ , $\gamma$ and $\delta$ are arbitrary constants with $\alpha \delta -\beta \gamma =1$ ; $\lambda _0$ , …, $\lambda _3$ and $\sigma$ are arbitrary constants with $\sigma \ne 0$ , and $\,f$ is an arbitrary solution of (2).

Proof. We should integrate the system (5) with $\bar{\tilde{\theta }}=\bar \theta =\bar \theta _0$ with respect to the parameter functions $T$ , $X$ , $Y$ and $U^1$ . The equations (5d) take the form

\begin{equation*} X=\dfrac {Y_t +xY_y}{T_t +xT_y},\quad X_x^2=T_t +xT_y. \end{equation*}

In view of the first of these equations, the second equation reduces to $(T_tY_y-T_yY_t)^2=(T_t+xT_y)^5$ , which implies $T_y=0$ , $T_t\gt 0$ and

\begin{equation*} Y=\varepsilon T_t^{3/2}y+Y^0(t), \end{equation*}

where $\varepsilon =\pm 1$ and $Y^0$ is a function of $t$ arising due to the integration with respect to $y$ . This is why we have $X=(Y_t +xY_y)/T_t$ , and hence $X_{xx}=0$ . Then the equation (5c) simplifies to $X_t+xX_y=-2X_xU^1_x/U^1$ and thus integrates to

\begin{equation*} U^1=V(t,y)\exp \left (-\frac {T_{tt}}{2T_t}x^2 -3\frac {2T_{ttt}T_t-T^2_{tt}}{8T_t^2}xy-\frac {(Y^0/T_t)_t}{2T_t^{5/2}}x \right ), \end{equation*}

where $V$ is a nonvanishing smooth function of $(t,y)$ , the explicit expression for which will be derived below. Substituting the expression for $U^1$ into the restricted equation (5b),

\begin{equation*} \left (\frac {U^1_x}{U^1}\right )^2-\left (\frac {U^1_x}{U^1}\right )_x+\frac {U^1_t+xU^1_y}{U^1}=0, \end{equation*}

leads to an equation whose left-hand side is a quadratic polynomial in $x$ . Collecting the coefficients of $x^2$ , we derive the equation $T_{ttt}/T_t-\frac 32(T_{tt}/T_t)^2=0$ , meaning that the Schwarzian derivative of $T$ is zero. Therefore, $T$ is a linear fractional function of $t$ , $T=(\alpha t+\beta )/(\gamma t+\delta )$ . Since the constants $\alpha$ , $\beta$ , $\gamma$ and $\delta$ are defined up to a constant nonzero multiplier and $T_t\gt 0$ , we can assume that $\alpha \delta -\beta \gamma =1$ . Then, the result of collecting the coefficients of $x$ in the above equation can be represented, up to an inessential multiplier, in the form

\begin{equation*} \frac {V_y}V=\frac {6\gamma ^3}{(\gamma t+\delta )^3}y+\left ((\gamma t+\delta )^3Y^0\right )_{ttt} -6\gamma ^2\left ((\gamma t+\delta )Y^0\right )_t\!. \end{equation*}

The general solution of the last equation as an equation with respect to $V$ is

\begin{equation*} V=\phi (t)\exp \left (\frac {3\gamma ^3}{(\gamma t+\delta )^3}y^2 +\left ((\gamma t+\delta )^3Y^0\right )_{ttt}y -6\gamma ^2\left ((\gamma t+\delta )Y^0\right )_ty\right )\!, \end{equation*}

where $\phi$ is a nonvanishing smooth function of $t$ . Analogously, we collect the summands without $x$ , substitute the above representation for $V$ into the obtained equation, split the result with respect to $y$ and in addition neglect inessential multipliers. This gives the system of two equations

\begin{align*} \left ((\gamma t+\delta )^3Y^0\right )_{tttt}=0, \quad \frac{\phi _t}{\phi }-\frac{2\gamma }{\gamma t+\delta }+\frac 1{4}(\gamma t+\delta )^4\left (\left ((\gamma t+\delta )Y^0\right )_{tt}\right )^2=0, \end{align*}

whose general solution can be represented as

\begin{align*} &Y^0=\dfrac{\lambda _3t^3+\lambda _2t^2+\lambda _1t+\lambda _0}{(\gamma t+\delta )^3},\\ &\phi =\sigma (\gamma t+\delta )^2\exp \left ( \frac{\gamma \psi _t^{\,2}}{\gamma t+\delta } -\frac{3\gamma ^2\psi _t\psi }{(\gamma t+\delta )^2} +\frac{3\gamma ^3\psi ^2}{(\gamma t+\delta )^3} \right ) \exp \left (-\lambda _2^2t-3\lambda _3\lambda _2t^2-3\lambda _3^2t^3\right ), \end{align*}

where $\psi$ is an arbitrary at most cubic polynomial of $t$ , $\psi (t)\,:\!=\lambda _3t^3+\lambda _2t^2+\lambda _1t+\lambda _0$ , and $\lambda _0$ , …, $\lambda _3$ and $\sigma$ are arbitrary constants with $\sigma \ne 0$ . Since the constant parameters $\alpha$ , $\beta$ , $\gamma$ and $\delta$ are still defined up to the multiplier $\pm 1$ , we can choose these parameters in such a way that $\varepsilon |\gamma t+\delta |=\gamma t+\delta$ and then neglect the parameter $\varepsilon$ .

Finally, the equation (5e) takes the form

\begin{equation*} \left (\dfrac {U^0}{U^1}\right )_t+x\left (\dfrac {U^0}{U^1}\right )_y=\left (\dfrac {U^0}{U^1}\right )_{xx}, \end{equation*}

and thus $U^0=U^1\,f$ , where $\,f=\,f(t,x,y)$ is an arbitrary solution of (2).

If the transformations of the form (6) and their compositions are properly interpreted, then the structure of the pseudogroup $G$ is simplified. Thus, the natural domain of a transformation $\Phi$ of the form (6) is

\begin{equation*}\textrm {dom}\,\Phi =(\textrm {dom}\,\,f\times \mathbb R_u)\setminus M_{\gamma \delta },\end{equation*}

that is, it can be assumed to coincide with the relative complement of the set $M_{\gamma \delta }\,:\!=\{(t,x,y,u)\in \mathbb R^4\mid \gamma t+\delta =0\}$ with respect to the set $\textrm{dom}\,\,f\times \mathbb R_u$ , where $\textrm{dom}\,\,f$ is the domain of the function $f$ . In view of the fact that parameters $\gamma$ and $\delta$ do not vanish simultaneously, the set $M_{\gamma \delta }$ is empty if $\gamma =0$ and is the hyperplane defined by the equation $t=-\delta/\gamma$ in the space $\mathbb R^4_{t,x,y,u}$ otherwise. We modify the standard definition of composition in the case of transformations of the form (6). More specifically, according to the standard definition, the domain of the composition $\Phi _1\circ \Phi _2=\!:\,\tilde \Phi$ of transformations $\Phi _1$ and $\Phi _2$ is the preimage of the domain of $\Phi _1$ with respect of $\Phi _2$ . If the transformations $\Phi _1$ and $\Phi _2$ are of the form (6), then $\textrm{dom}\,\tilde \Phi =\Phi _2^{-1}(\textrm{dom}\,\Phi _1)=(\textrm{dom}\,\,\tilde f\times \mathbb R_u) \setminus (M_{\gamma _2\delta _2}\cup M_{\tilde \gamma \tilde \delta })$ , where $\tilde \gamma =\gamma _1\alpha _2+\delta _1\gamma _2$ , $\tilde \delta =\gamma _1\beta _2+\delta _1\delta _2$ , $\textrm{dom}\,\,\tilde f= ((\pi _*\Phi _2)^{-1}\textrm{dom}\,\,f^1 )\cap \textrm{dom}\,\,f^2$ , $\pi$ is the natural projection onto $\mathbb R^3_{t,x,y}$ in $\mathbb R^4_{t,x,y,u}$ , and the parameters with indices 1 and 2 and tildes correspond to $\Phi _1$ , $\Phi _2$ and $\tilde \Phi$ , respectively. As the modified composition $\Phi _1\circ ^{\textrm{m}}\Phi _2$ of transformations $\Phi _1$ and $\Phi _2$ , we take the extension of $\Phi _1\circ \Phi _2$ by continuity to the set

\begin{equation*}\smash {\mathop {\textrm {dom}}\nolimits ^{\textrm {m}}\tilde \Phi \,:\!=(\textrm {dom}\,\,\tilde f\times \mathbb R_u)\setminus M_{\tilde \gamma \tilde \delta }},\end{equation*}

that is, $\textrm{dom}(\Phi _1\circ ^{\textrm{m}}\Phi _2)=\mathop{\textrm{dom}}\nolimits ^{\textrm{m}}\tilde \Phi$ . Therefore, $\Phi _1\circ ^{\textrm{m}}\Phi _2$ is the transformation of the form (6) with the same parameters as in $\Phi _1\circ \Phi _2$ and with its natural domain. This means that $\mathop{\textrm{dom}}\nolimits ^{\textrm{m}}\tilde \Phi =\textrm{dom}\,\tilde \Phi$ and the extension is trivial if $\gamma _1\gamma _2=0$ , and otherwise we redefine $\Phi _1\circ \Phi _2$ on the set $(\textrm{dom}\,\,\tilde f\times \mathbb R_u)\cap M_{\gamma _2\delta _2}$ .

Now we can analyse the structure of $G$ . The point transformations of the form

\begin{equation*} {\mathscr{Z}}(\,f)\,:\, \quad \tilde{t}=t,\quad \tilde{x}=x,\quad \tilde{y}=y,\quad \tilde{u}=u+\,f(t,x,y), \end{equation*}

where the parameter function $\,f=\,f(t,x,y)$ is an arbitrary solution of the equation (2), are associated with the linear superposition of solutions of this equation and thus can be considered as trivial. They constitute the normal pseudosubgroup $G^{\textrm{lin}}$ of the pseudogroup $G$ . The pseudogroup $G$ splits over $G^{\textrm{lin}}$ , $G=G^{\textrm{ess}}\ltimes G^{\textrm{lin}}$ , where $G^{\textrm{ess}}$ is the subgroup of $G$ consisting of the transformations of the form (6) with $\,f=0$ and with their natural domains, and thus it is an eight-dimensional Lie group. The Lie group structure of $G^{\textrm{ess}}$ is the main benefit of redefining the transformation composition above.Footnote ³ We call the subgroup $G^{\textrm{ess}}$ the essential point symmetry group of the equation (2).

The subgroup $G^{\textrm{ess}}$ itself splits over $R$ , $G^{\textrm{ess}}=F\ltimes R$ . Here $R$ and $F$ are the normal subgroup and the subgroup of $G^{\textrm{ess}}$ that are singled out by the constraints $\alpha =\delta =1$ , $\beta =\gamma =0$ and $\lambda _3=\lambda _2=\lambda _1=\lambda _0=0$ , $\sigma =1$ , respectively. They are isomorphic to the groups $\textrm{H}(2,\mathbb R)\times \mathbb Z_2$ and $\textrm{SL}(2,\mathbb R)$ , and their Lie algebras coincide with $\mathfrak r\simeq \textrm{h}(2,\mathbb R)$ and $\mathfrak f\simeq \textrm{sl}(2,\mathbb R)$ . Here $\textrm{H}(2,\mathbb R)$ denotes the rank-two real Heisenberg group. The normal subgroups $R_{\textrm{c}}$ and $R_{\textrm{d}}$ of $R$ that are isomorphic to $\textrm{H}(2,\mathbb R)$ and $\mathbb Z_2$ are constituted by the elements of $R$ with parameter values satisfying the constraints $\sigma \gt 0$ and $\lambda _3=\lambda _2=\lambda _1=\lambda _0=0$ , $\sigma \in \{-1,1\}$ , respectively. The isomorphisms of $F$ to $\textrm{SL}(2,\mathbb R)$ and of $R_{\textrm{c}}$ to $\textrm{H}(2,\mathbb R)$ are established by the correspondences

\begin{equation*} (\alpha,\beta,\gamma,\delta )_{\alpha \delta -\beta \gamma =1}\mapsto \begin {pmatrix} \alpha & \beta \\ \gamma & \delta \end {pmatrix}, \quad (\lambda _3,\lambda _2,\lambda _1,\lambda _0,\sigma )_{,\;\sigma \gt 0}\mapsto \begin {pmatrix} 1 &\quad 3\lambda _3 &\quad -\lambda _2 &\quad \ln \sigma \\ 0 & 1 & 0 & \lambda _0\\ 0 & 0 & 1 & \lambda _1\\ 0 & 0 & 0 & 1 \end {pmatrix}. \end{equation*}

Thus, $F$ and $R_{\textrm{c}}$ are connected subgroups of $G^{\textrm{ess}}$ , but $R_{\textrm{d}}$ is not. The natural conjugacy action of the group $F$ on the normal subgroup $R$ is given by $(\tilde \lambda _3,\tilde \lambda _2,\tilde \lambda _1,\tilde \lambda _0,\tilde \sigma ) =(\lambda _3,\lambda _2,\lambda _1,\lambda _0,\sigma )\,A$ in the parameterisation (6) of $G$ , where $A=\varrho _3(\alpha,\beta,\gamma,\delta )\oplus (1)$ , and $\varrho _3$ is the standard real irreducible four-dimensional representation of $\textrm{SL}(2,\mathbb R)$ , which can be identified with the action of $\textrm{SL}(2,\mathbb R)$ on binary cubics,

\begin{equation*} \varrho _3\,:\, \ (\alpha,\beta,\gamma,\delta )_{\alpha \delta -\beta \gamma =1}\mapsto \begin {pmatrix} \alpha ^3 & 3\alpha ^2\beta & 3\alpha \beta ^2 & \quad\beta ^3\\ \alpha ^2\gamma &\quad 2\alpha \beta \gamma +\alpha ^2\delta &\quad 2\alpha \beta \delta +\beta ^2\gamma &\quad \beta ^2\delta \\ \alpha \gamma ^2 & \quad 2\alpha \gamma \delta +\beta \gamma ^2 & \quad 2\beta \gamma \delta +\alpha \delta ^2 & \quad\beta \delta ^2\\ \gamma ^3 & 3\gamma ^2\delta & 3\gamma \delta ^2 & \quad\delta ^3 \end {pmatrix}. \end{equation*}

Summing up, the group $G^{\textrm{ess}}$ is isomorphic to $ (\textrm{SL}(2,\mathbb R)\ltimes _{\varphi }\textrm{H}(2,\mathbb R) )\times \mathbb Z_2$ , where the antihomomorphism $\varphi \,:\, \textrm{SL}(2,\mathbb R)\to \textrm{Aut}(\textrm{H}(2,\mathbb R))$ is defined, in the chosen local coordinates, by $ \varphi (\alpha,\beta,\gamma,\delta )=(\lambda _3,\lambda _2,\lambda _1,\lambda _0,\sigma ) \mapsto (\lambda _3,\lambda _2,\lambda _1,\lambda _0,\sigma )A.$

Transformations from the one-parameter subgroups of $G^{\textrm{ess}}$ that are generated by the basis elements of $\mathfrak{g}^{\textrm{ess}}$ given in (4) are of the following form:

\begin{equation*} \begin{array}{l@{\kern+10pt}l@{\kern+10pt}l@{\kern+10pt}l@{\kern+10pt}l} \mathscr P^t(\epsilon )\,:\, & \tilde t=t+\epsilon, & \tilde x=x, & \tilde y=y, & \tilde u=u,\\ \mathscr D (\epsilon )\,:\, & \tilde t=\textrm{e}^{2\epsilon }t, & \tilde x=\textrm{e}^\epsilon x, & \tilde y=\textrm{e}^{3\epsilon }x, & \tilde u=\textrm{e}^{-2\epsilon }u,\\ \mathscr K(\epsilon ) \,:\, & \tilde t=\dfrac t{1\!-\!\epsilon t}, & \tilde x=\dfrac x{1\!-\!\epsilon t}+\dfrac{3\epsilon y}{(1\!-\!\epsilon t)^2}, & \tilde y=\dfrac y{(1\!-\!\epsilon t)^3}, &\tilde u=(1\!-\!\epsilon t)^2\textrm{e}^{\frac{\epsilon x^2}{1-\epsilon t}+\frac{3\epsilon ^2xy}{(1-\epsilon t)^2}+\frac{3\epsilon ^3y^2}{(1-\epsilon t)^3}}u,\\ \mathscr P^3(\epsilon )\,:\, & \tilde t=t, & \tilde x=x+3\epsilon t^2,\ \ & \tilde y=y+\epsilon t^3,\ \ & \tilde u=\textrm{e}^{3\epsilon (y-tx)-3\epsilon ^2t^3}u,\\ \mathscr P^2(\epsilon )\,:\, & \tilde t=t, & \tilde x=x+2\epsilon t, & \tilde y=y+\epsilon t^2, & \tilde u=\textrm{e}^{-\epsilon x+\epsilon ^2t}u,\\ \mathscr P^1(\epsilon )\,:\, \ & \tilde t=t, & \tilde x=x+\epsilon, & \tilde y=y+\epsilon t, & \tilde u=u,\\ \mathscr P^0(\epsilon )\,:\, & \tilde t=t, & \tilde x=x, & \tilde y=y+\epsilon, & \tilde u=u,\\ \mathscr I (\epsilon )\,:\, & \tilde t=t, & \tilde x=x, & \tilde y=y,\ & \tilde u=\textrm{e}^\epsilon u,\\ \end{array} \end{equation*}

where $\epsilon$ is the group parameter. At the same time, using this basis of $\mathfrak{g}^{\textrm{ess}}$ in the course of studying the structure of the group $G^{\textrm{ess}}$ hides some of its important properties and complicates its study.

Although the pushforward of the group $G^{\textrm{ess}}$ by the natural projection of $\mathbb R^4_{t,x,y,u}$ onto $\mathbb R_t$ coincides with the group of linear fractional transformations of $t$ and is thus isomorphic to the group $\textrm{PSL}(2,\mathbb R)$ , the subgroup $F$ of $G^{\textrm{ess}}$ is isomorphic to the group $\textrm{SL}(2,\mathbb R)$ , and its Iwasawa decomposition is given by the one-parameter subgroups of $G^{\textrm{ess}}$ , respectively, generated by the vector fields $\mathcal{P}^t+\mathcal{K}$ , $\mathcal{D}$ and $\mathcal{P}^t$ . The first subgroup, which is associated with $\mathcal{P}^t+\mathcal{K}$ , consists of the point transformations

(7) \begin{align} \begin{split}& \tilde t=\frac{t\cos \epsilon -\sin \epsilon }{t\sin \epsilon +\cos \epsilon }, \quad \tilde x=\frac x{t\sin \epsilon +\cos \epsilon } -\frac{3y\sin \epsilon }{(t\sin \epsilon +\cos \epsilon )^2}, \quad \tilde y=\frac y{(t\sin \epsilon +\cos \epsilon )^3}, \\[1ex]& \tilde u=(t\sin \epsilon +\cos \epsilon )^2\exp \left ( \frac{x^2\sin \epsilon }{t\sin \epsilon +\cos \epsilon } -\frac{3xy\sin ^2\epsilon }{(t\sin \epsilon +\cos \epsilon )^2} +\frac{3y^2\sin ^3\epsilon }{(t\sin \epsilon +\cos \epsilon )^3} \right )u \end{split} \end{align}

parameterised by an arbitrary constant $\epsilon$ , which is defined by the corresponding transformation up to a summand $2\pi k$ , $k\in \mathbb Z$ .

The equation (2) is invariant with respect to the involution $\mathscr J$ simultaneously alternating the sign of $(x,y)$ ,

\begin{equation*}\mathscr J \,:\, (t,x,y,u)\mapsto (t,-x,-y,u).\end{equation*}

In the context of the one-parameter subgroups of $G^{\textrm{ess}}$ that are generated by the basis elements of $\mathfrak{g}^{\textrm{ess}}$ listed in (4), the involution $\mathscr J$ looks like a discrete point symmetry transformation of (2), but in fact this is not the case. It belongs to the one-parameter subgroup of $G^{\textrm{ess}}$ generated by $\mathcal{P}^t+\mathcal{K}$ . More precisely, it coincides with the transformation (7) with $\epsilon =\pi$ . The value $\epsilon =\pi/2$ corresponds to the transformation

\begin{equation*} \mathscr K'\,:\, \ \ \tilde t=-\dfrac 1t,\ \ \tilde x=\dfrac xt-3\dfrac y{t^2},\ \ \tilde y=\dfrac y{t^3},\ \ \tilde u=t^2\textrm {e}^{\frac {x^2}t-\frac {3xy}{t^2}+\frac {3y^2}{t^3}}u, \end{equation*}

which also deceptively looks, in the above context, like a discrete point symmetry transformation of (2) being independent with $\mathscr J$ since the factorisation $\mathscr J\circ \mathscr K'=\mathscr P^t(1)\circ \mathscr K(1)\circ \mathscr P^t(1)$ is not intuitive. This is why it is relevant to accurately describe discrete point symmetries of the equation (2).

Corollary 3. A complete list of discrete point symmetry transformations of the remarkable Fokker–Planck equation (2) that are independent up to combining with each other and with continuous point symmetry transformations of this equation is exhausted by the single involution $\mathscr I'$ alternating the sign of $u$ ,

\begin{equation*}\mathscr I'\,:\, (t,x,y,u)\mapsto (t,x,y,-u).\end{equation*}

Thus, the quotient group of the complete point symmetry pseudogroup $G$ with respect to its identity component is isomorphic to $\mathbb Z_2$ .

Hereafter, the transformations $\mathscr P^t(\epsilon )$ , $\mathscr D(\epsilon )$ , $\mathscr K(\epsilon )$ , $\mathscr P^3(\epsilon )$ , $\mathscr P^2(\epsilon )$ , $\mathscr P^1(\epsilon )$ , $\mathscr P^0(\epsilon )$ , $\mathscr I(\epsilon )$ , $\mathscr J$ , $\mathscr K'$ , $\mathscr J'$ and $\mathscr Z(\,f)$ are called elementary point symmetry transformations of the equation (2).

In view of Theorem 2, the formal application of the point symmetry transformation $\Phi \,:\!=\mathscr I\left(\!\ln \frac{\sqrt 3}{2\pi }\right)\circ \mathscr P^0(y')\circ \mathscr P^1(x')\circ \mathscr P^t(t')\circ \mathscr K'$ of the equation (2),

(8) \begin{align} \begin{split} \Phi \,:\, \quad \tilde t&=-\frac 1t+t',\quad \tilde x=\frac xt-3\frac y{t^2}+x',\quad \tilde y=\frac y{t^3}+y', \\ \tilde u &=\frac{\sqrt 3}{2\pi }t^2 \exp \left (\frac{x^2}t-3\frac{xy}{t^2}+3\frac{y^2}{t^3}+3x'x-3x'\frac yt+3(x')^2t\right ) u, \end{split} \end{align}

maps the function $u(t,x,y)=1-H(t)$ to the fundamental solution (3) of the equation (2). Recall that $H$ denotes the Heaviside step function. Note that attempts to interpret this fundamental solution within the framework of group analysis of differential equations were made in [Reference Güngör17, Reference Kovalenko, Kopas and Stogniy20].

4. Classification of inequivalent subalgebras

In order to carry out the Lie reductions of codimension one and two for the equation (2) in the optimal way, we need to classify one- and two-dimensional subalgebras of $\mathfrak{g}^{\textrm{ess}}$ up to the $G^{\textrm{ess}}$ -equivalence. In the course of this classification, we use the Levi decomposition $\mathfrak{g}^{\textrm{ess}}=\mathfrak f\mathbin{\mbox{${\,\,\rule{.4pt}{1.07ex}}{\!\!\in }$}}\mathfrak r$ and the fact that the Levi factor $\mathfrak f$ is isomorphic to $\textrm{sl}(2,\mathbb R)$ . This allows us to apply the technique of classifying the subalgebras of an algebra with a proper ideal suggested in [Reference Patera, Winternitz and Zassenhaus29]. This technique becomes simpler if the algebra under consideration can be represented as the semidirect sum of a subalgebra and an ideal, which are $\mathfrak f$ and $\mathfrak r$ for the algebra $\mathfrak{g}^{\textrm{ess}}$ , respectively. An additional simplification is that an optimal list of subalgebras of $\textrm{sl}(2,\mathbb R)$ is well known (see, e.g. [Reference Patera and Winternitz28, Reference Popovych, Boyko, Nesterenko and Lutfullin32]). Thus, for the realisation $\mathfrak f$ of $\textrm{sl}(2,\mathbb R)$ , this list consists of the subalgebras $\{0\}$ , $\langle \mathcal{P}^t\rangle$ , $\langle \mathcal{D}\rangle$ , $\langle \mathcal{P}^t+\mathcal{K}\rangle$ , $\langle \mathcal{P}^t, \mathcal{D}\rangle$ and $\mathfrak f$ itself. The subalgebras $\mathfrak s_1$ and $\mathfrak s_2$ of $\mathfrak{g}^{\textrm{ess}}$ are definitely $G^{\textrm{ess}}$ -inequivalent if their projections $\pi _{\mathfrak f}\mathfrak s_1$ and $\pi _{\mathfrak f}\mathfrak s_2$ are $F$ -inequivalent, see Section 3. Here and in what follows, $\pi _{\mathfrak f}$ and $\pi _{\mathfrak r}$ denote the natural projections of $\mathfrak{g}^{\textrm{ess}}$ onto $\mathfrak f$ and $\mathfrak r$ according to the decomposition $\mathfrak{g}^{\textrm{ess}}=\mathfrak f\dotplus \mathfrak r$ of the vector space $\mathfrak{g}^{\textrm{ess}}$ as the direct sum of its subspaces $\mathfrak f$ and $\mathfrak r$ . In other words, we can use the projections $\pi _{\mathfrak f}\mathfrak s$ of the subalgebras $\mathfrak s$ of $\mathfrak{g}^{\textrm{ess}}$ of the same dimension for partitioning the set of these subalgebras into subsets such that each subset contains no subalgebra being equivalent to a subalgebra from another subset.

Let us specifically describe applying the above technique to the classification of one- and two-dimensional subalgebras of $\mathfrak{g}^{\textrm{ess}}$ up to the $G^{\textrm{ess}}$ -equivalence. We fix the dimension $d$ of subalgebras $\mathfrak s$ to be classified, either $d=1$ or $d=2$ , and consider the subalgebras $\mathfrak s_{\mathfrak f}$ of $\mathfrak f$ from the above list with dimension $d'$ less than or equal to $d$ . For each of these subalgebras, we take the set of $d$ -dimensional subalgebras of $\mathfrak{g}^{\textrm{ess}}$ with $\pi _{\mathfrak f}\mathfrak s=\mathfrak s_{\mathfrak f}$ and construct a complete list of $G^{\textrm{ess}}$ -inequivalent subalgebras in this set.

One usually classifies subalgebras of a Lie algebra up to their equivalence generated by the group of inner automorphisms of this algebra, and this group is computed by summing up Lie series or solving Cauchy problems with the adjoint action of the algebra on itself, see, for example, [Reference Olver25, Section 3.3]. At the same time, we already know that the algebra $\mathfrak{g}^{\textrm{ess}}$ is the Lie algebra of the group $G^{\textrm{ess}}$ , and the actions of $G^{\textrm{ess}}$ and of the group of inner automorphisms of $\mathfrak{g}^{\textrm{ess}}$ on $\mathfrak{g}^{\textrm{ess}}$ coincide. Moreover, recall that for the purpose of finding Lie invariant solutions of the equation (2) subalgebras of $\mathfrak{g}^{\textrm{ess}}$ have in fact to be classified up to the $G^{\textrm{ess}}$ -equivalence. This is why following, for example, [Reference Dos Santos Cardoso-Bihlo, Bihlo and Popovych15] we directly compute the action of $G^{\textrm{ess}}$ on $\mathfrak{g}^{\textrm{ess}}$ by pushing forward vector fields from $\mathfrak{g}^{\textrm{ess}}$ by transformations from $G^{\textrm{ess}}$ . The nonidentity pushforwards of basis elements of $\mathfrak{g}^{\textrm{ess}}$ by the elementary transformations from $G^{\textrm{ess}}$ are the following:

\begin{align*} \begin{array}{l}\begin{array}{l@{\kern+12pt}l} \mathscr P^t(\epsilon )_*\mathcal{D} =\mathcal{D}-2\epsilon \mathcal{P}^t, & \mathscr K(\epsilon )_*\mathcal{D} =\mathcal{D}+2\epsilon \mathcal{K}, \\\mathscr P^t(\epsilon )_*\mathcal{K} =\mathcal{K}-\epsilon \mathcal{D}+\epsilon ^2\mathcal{P}^t, & \mathscr K(\epsilon )_*\mathcal{P}^t=\mathcal{P}^t+\epsilon \mathcal{D}+\epsilon ^2\mathcal{K}, \\\mathscr P^t(\epsilon )_*\mathcal{P}^3=\mathcal{P}^3-3\epsilon \mathcal{P}^2+3\epsilon ^2\mathcal{P}^1-\epsilon ^3\mathcal{P}^0, & \mathscr K(\epsilon )_*\mathcal{P}^2=\mathcal{P}^2+\epsilon \mathcal{P}^3, \\ \mathscr P^t(\epsilon )_*\mathcal{P}^2=\mathcal{P}^2-2\epsilon \mathcal{P}^1+\epsilon ^2\mathcal{P}^0, & \mathscr K(\epsilon )_*\mathcal{P}^1=\mathcal{P}^1+2\epsilon \mathcal{P}^2+\epsilon ^2\mathcal{P}^3, \\\mathscr P^t(\epsilon )_*\mathcal{P}^1=\mathcal{P}^1-\epsilon \mathcal{P}^0, & \mathscr K(\epsilon )_*\mathcal{P}^0=\mathcal{P}^0+3\epsilon \mathcal{P}^1+3\epsilon ^2\mathcal{P}^2+\epsilon ^3\mathcal{P}^3,\end{array} \\[30pt] \begin{array}{l@{\kern+12pt}l@{\kern+12pt}l} \mathscr D(\epsilon )_*\mathcal{P}^t=e^{2\epsilon }\mathcal{P}^t, &\mathscr D(\epsilon )_*\mathcal{P}^3=e^{-3\epsilon }\mathcal{P}^3, & \mathscr D(\epsilon )_*\mathcal{P}^1=e^{\epsilon }\mathcal{P}^1, \\ \mathscr D(\epsilon )_*\mathcal{K} =e^{-2\epsilon }\mathcal{K}, & \mathscr D(\epsilon )_*\mathcal{P}^2=e^{-\epsilon }\mathcal{P}^2, & \mathscr D(\epsilon )_*\mathcal{P}^0=e^{3\epsilon }\mathcal{P}^0,\\\end{array}\\[20pt] \begin{array}{l@{\kern+12pt}l} \mathscr P^3(\epsilon )_*\mathcal{P}^t=\mathcal{P}^t+3\epsilon \mathcal{P}^2, & \mathscr P^0(\epsilon )_*\mathcal{D}=\mathcal{D}-3\epsilon \mathcal{P}^0,\\\mathscr P^3(\epsilon )_*\mathcal{D} =\mathcal{D}+3\epsilon \mathcal{P}^3, & \mathscr P^0(\epsilon )_*\mathcal{K}=\mathcal{K}-3\epsilon \mathcal{P}^1,\\\mathscr P^3(\epsilon )_*\mathcal{P}^0=\mathcal{P}^0+3\epsilon \mathcal{I}, & \mathscr P^0(\epsilon )_*\mathcal{P}^3=\mathcal{P}^3-3\epsilon \mathcal{I},\\ \end{array} \\[20pt] \begin{array}{l@{\kern+12pt}l}\mathscr P^2(\epsilon )_*\mathcal{P}^t=\mathcal{P}^t+2\epsilon \mathcal{P}^1-\epsilon ^2\mathcal{I}, &\mathscr P^1(\epsilon )_*\mathcal{P}^t=\mathcal{P}^t+\epsilon \mathcal{P}^0,\\\mathscr P^2(\epsilon )_*\mathcal{D} =\mathcal{D}+\epsilon \mathcal{P}^2, & \mathscr P^1(\epsilon )_*\mathcal{D} =\mathcal{D}-\epsilon \mathcal{P}^1,\\\mathscr P^2(\epsilon )_*\mathcal{K} =\mathcal{K}-\epsilon \mathcal{P}^3, & \mathscr P^1(\epsilon )_*\mathcal{K} =\mathcal{K}-2\epsilon \mathcal{P}^2-\epsilon ^2\mathcal{I},\\\mathscr P^2(\epsilon )_*\mathcal{P}^1=\mathcal{P}^1-\epsilon \mathcal{I}, & \mathscr P^1(\epsilon )_*\mathcal{P}^2=\mathcal{P}^2+\epsilon \mathcal{I},\\ \end{array} \\[30pt] \begin{array}{l} \mathscr J_*(\mathcal{P}^3,\mathcal{P}^2,\mathcal{P}^1,\mathcal{P}^0)=({-}\mathcal{P}^3,-\mathcal{P}^2,-\mathcal{P}^1,-\mathcal{P}^0),\\ \mathscr {{K_*}^{\!\!\prime}}(\mathcal{P}^t,\mathcal{D},\mathcal{K},\mathcal{P}^3,\mathcal{P}^2,\mathcal{P}^1,\mathcal{P}^0)=(\mathcal{K},-\mathcal{D},\mathcal{P}^t,\mathcal{P}^0,-\mathcal{P}^1,\mathcal{P}^2,-\mathcal{P}^3). \end{array} \end{array} \end{align*}

The result of the classification of one- and two-dimensional subalgebras of $\mathfrak{g}^{\textrm{ess}}$ up to the $G^{\textrm{ess}}$ -equivalence is presented in the two subsequent lemmas.

Lemma 4. A complete list of $G^{\textit{ess}}$ -inequivalent one-dimensional subalgebras of $\mathfrak{g}^{\textit{ess}}$ is exhausted by the subalgebras

\begin{align*} &\mathfrak s_{1.1}=\langle \mathcal{P}^t+\mathcal{P}^3\rangle,\ \ \mathfrak s_{1.2}^\delta =\langle \mathcal{P}^t+\delta \mathcal{I}\rangle,\ \ \mathfrak s_{1.3}^\nu =\langle \mathcal{D}+\nu \mathcal{I}\rangle,\ \ \mathfrak s_{1.4}^\mu =\langle \mathcal{P}^t+\mathcal{K}+\mu \mathcal{I}\rangle,\\ &\mathfrak s_{1.5}^\varepsilon =\langle \mathcal{P}^2+\varepsilon \mathcal{P}^0\rangle,\ \ \mathfrak s_{1.6}=\langle \mathcal{P}^1\rangle,\ \ \mathfrak s_{1.7}=\langle \mathcal{P}^0\rangle,\ \ \mathfrak s_{1.8}=\langle \mathcal{I}\rangle, \end{align*}

where $\varepsilon \in \{-1,1\}$ , $\delta \in \{-1,0,1\}$ , and $\mu$ and $\nu$ are arbitrary real constants with $\nu \geqslant 0$ .

Proof. A complete set of $F$ -inequivalent $d'$ -dimensional subalgebras of $\mathfrak f$ with $d'\leqslant 1$ consists of the subalgebras $\langle \mathcal{P}^t\rangle$ , $\langle \mathcal{D}\rangle$ , $\langle \mathcal{P}^t+\mathcal{K}\rangle$ and $\{0\}$ . Therefore, without loss of generality we can consider only one-dimensional subalgebras of $\mathfrak{g}^{\textrm{ess}}$ with basis vector fields $Q$ of the general form

\begin{equation*} Q = \hat Q+a_3\mathcal {P}^3+3a_2\mathcal {P}^2+3a_1\mathcal {P}^1+a_0\mathcal {P}^0+b\mathcal {I}, \end{equation*}

where $\hat Q\in \{\mathcal{P}^t,\mathcal{D},P^t+\mathcal{K},0\}$ and $a_0$ , …, $a_3$ and $b$ are real constants. We factor out the multiplier 3 in the coefficients of $\mathcal{P}^1$ and $\mathcal{P}^2$ since then the coefficients $a_3$ , $a_2$ , $a_1$ and $a_0$ are changed under the action of $F$ in the same way as the coefficients of the real cubic binary form $a_3x^3+3a_2x^2y+3a_1xy^2+a_0y^3$ are changed under the standard action of the group of linear fractional transformations on such forms, cf. [Reference Olver24, Example 2.22]. We need to further simplify the vector fields $Q$ of the above form by acting with elements of $G^{\textrm{ess}}$ on them and/or scaling them.

Let $\hat Q=\mathcal{P}^t$ . If $a_3\ne 0$ , then we successively set $a_3=1$ , $a_2=a_1=a_0=0$ and $b=0$ using $\mathscr D(e^{\epsilon _4})_*$ simultaneously with rescaling $Q$ , $\mathscr P^1(\epsilon _1)_*\circ \mathscr P^2(\epsilon _2)_*\circ \mathscr P^3(\epsilon _3)_*$ and $\mathscr P^0(\epsilon _0)_*$ , with appropriate constants $\epsilon _i$ , $i=0,\dots,4$ , respectively. This gives the subalgebra $\mathfrak s_{1.1}$ . Similarly, for $a_3=0$ , we apply $\mathscr P^3(\epsilon _2)_*$ , $\mathscr P^2(\epsilon _1)_*$ and $\mathscr P^1(\epsilon _0)_*$ with appropriate constants $\epsilon _i$ , $i=1,2,3$ , to get rid of $a_2$ , $a_1$ and $a_0$ . Then, the action of $\mathscr D(e^{\epsilon _b})_*$ with appropriate $\epsilon _b$ and rescaling $Q$ allow us to set $b\in \{-1,1\}$ if $b\ne 0$ . Thus, we obtain the subalge- bras $\mathfrak s_{1.2}^\delta$ .

For $\hat Q=\mathcal{D}$ and $\hat Q=\mathcal{P}^t+\mathcal{K}$ , removing the summands with $\mathcal{P}^3$ , $\mathcal{P}^2$ , $\mathcal{P}^1$ and $\mathcal{P}^0$ is analogous. The only possible further simplification is that in the case $\hat Q=\mathcal{D}$ , the sign of the obtained value of $b$ can be alternated using $\mathscr K_*^{\prime}$ and alternating the sign of $Q$ . This results in the collections of the subalgebras $\mathfrak s_{1.3}^\nu$ and $\mathfrak s_{1.4}^\mu$ , respectively.

Let $\hat Q=0$ . The classification of real binary cubics presented in the second table on page 26 in [Reference Olver24] implies that, up to the $F$ -equivalence and rescaling $Q$ , we have $a_3=0$ and $(3a_2,3a_1,a_0)\in \{(1,\varepsilon,0),(0,1,0),(0,0,1),(0,0,0)\}$ . If $(a_3,a_2,a_1,a_0)\ne (0,0,0,0)$ , then using one of the actions $\mathscr P^3(\epsilon )_*$ , $\mathscr P^2(\epsilon )_*$ , $\mathscr P^1(\epsilon )_*$ and $\mathscr P^0(\epsilon )_*$ we can set $b=0$ ; otherwise, we set $b=1$ by rescaling $Q$ . Thus, we obtain the rest of subalgebras from the list given in the lemma’s statement.

Lemma 5. A complete list of $G^{\textit{ess}}$ -inequivalent two-dimensional subalgebras of $\mathfrak{g}^{\textit{ess}}$ is given by

\begin{align*} &\mathfrak s_{2.1}^\mu =\langle \mathcal{P}^t,\mathcal{D}+\mu \mathcal{I}\rangle, \ \ \mathfrak s_{2.2}^\delta =\langle \mathcal{P}^t+\delta \mathcal{I},\mathcal{P}^0\rangle,\ \ \mathfrak s_{2.3}=\langle \mathcal{P}^t,\mathcal{P}^0+\mathcal{I}\rangle,\\ &\mathfrak s_{2.4}^\mu =\langle \mathcal{D}+\mu \mathcal{I},\mathcal{P}^1\rangle,\ \ \mathfrak s_{2.5}^\mu =\langle \mathcal{D}+\mu \mathcal{I},\mathcal{P}^0\rangle,\ \ \\ &\mathfrak s_{2.6}=\langle \mathcal{P}^0,\mathcal{P}^1\rangle,\ \ \mathfrak s_{2.7}=\langle \mathcal{P}^0,\mathcal{P}^2\rangle,\ \ \mathfrak s_{2.8}^\varepsilon =\langle \mathcal{P}^1,\mathcal{P}^3+\varepsilon \mathcal{P}^0\rangle,\ \ \\ &\mathfrak s_{2.9}=\langle \mathcal{P}^t+\mathcal{P}^3,\mathcal{I}\rangle,\ \ \mathfrak s_{2.10}^\delta =\langle \mathcal{P}^t,\mathcal{I}\rangle,\ \ \mathfrak s_{2.11}=\langle \mathcal{D},\mathcal{I}\rangle,\ \ \mathfrak s_{2.12}=\langle \mathcal{P}^t+\mathcal{K},\mathcal{I}\rangle, \\ &\mathfrak s_{2.13}^\varepsilon =\langle \mathcal{P}^2+\varepsilon \mathcal{P}^0,\mathcal{I}\rangle,\ \ \mathfrak s_{2.14}=\langle \mathcal{P}^1,\mathcal{I}\rangle,\ \ \mathfrak s_{2.15}=\langle \mathcal{P}^0,\mathcal{I}\rangle, \end{align*}

where $\varepsilon \in \{-1,1\}$ , $\delta \in \{-1,0,1\}$ , and $\mu$ is an arbitrary real constant.

Proof. A complete set of $F$ -inequivalent $d'$ -dimensional subalgebras of $\mathfrak f$ with $d'\leqslant 2$ consists of the subalgebras $\langle \mathcal{P}^t,\mathcal{D}\rangle$ , $\langle \mathcal{P}^t\rangle$ , $\langle \mathcal{D}\rangle$ , $\langle \mathcal{P}^t+\mathcal{K}\rangle$ and $\{0\}$ . Therefore, without loss of generality, we can consider only two-dimensional subalgebras of $\mathfrak{g}^{\textrm{ess}}$ with basis vector fields of the general form $Q_i=\hat Q_i+\check Q_i$ , $i=1,2$ , where

\begin{equation*} (\hat Q_1,\hat Q_2)\in \big \{(\mathcal {P}^t,\mathcal {D}),\,(\mathcal {P}^t,0),\,(\mathcal {D},0),\,(\mathcal {P}^t+\mathcal {K},0),\,(0,0)\big \}, \end{equation*}

and $\check Q_i=a_{i3}\mathcal{P}^3+3a_{i2}\mathcal{P}^2+3a_{i1}\mathcal{P}^1+a_{i0}\mathcal{P}^0+b_i\mathcal{I}$ with real constants $a_{ij}$ and $b_i$ , $i=1,2$ , $j=0,\dots,3$ . We further need to simplify $\check Q_i$ by linearly recombining the vector fields $Q_i$ and acting on them by elements from $G^{\textrm{ess}}$ , while simultaneously maintaining the closedness of the subalgebras with respect to the Lie bracket, $[Q_1,Q_2]\in \langle Q_1,Q_2\rangle$ , under the condition $\dim \langle Q_1,Q_2\rangle =2$ and preserving $\hat Q_i$ .

We separately consider each of the values of $(\hat Q_1,\hat Q_2)$ in the above set.

$\boldsymbol{(\mathcal{P}^t,\mathcal{D}).}$ Then according to Lemma 4, up to the $G^{\textrm{ess}}$ -equivalence, the vector field $Q_1$ is equal to either $\mathcal{P}^t+\mathcal{P}^3$ or $\mathcal{P}^t+\delta \mathcal{I}$ . The first value is not appropriate, since then $[Q_1,Q_2]=2\mathcal{P}^t-3\mathcal{P}^3+\tilde Q$ with $\tilde Q\in \langle \mathcal{P}^2,\mathcal{P}^1,\mathcal{P}^0,\mathcal{I}\rangle$ and thus $[Q_1,Q_2]\notin \langle Q_1,Q_2\rangle$ for any $\check Q_2\in \mathfrak r$ . For the second value, requiring the condition $[Q_1,Q_2]\in \langle Q_1,Q_2\rangle$ implies $a_{23}=a_{22}=a_{21}=\delta =0$ . Acting by $\mathscr P^0(a_{20})_*$ on $Q_2$ , we can set $a_{20}=0$ . Thus, we obtain the subalgebra $\mathfrak s_{2.1}^\mu$ with $\mu =b_2$ .

In all the other cases, we have $\hat Q_2=0$ . If, in addition, $a_{2j}=0$ , $j=0,\dots,3$ , then $Q_2=\mathcal{I}$ , and thus, the basis elements $Q_1$ and $Q_2$ commute. The corresponding canonical forms of $Q^1$ modulo the $G^{\textrm{ess}}$ -equivalence are given by the basis elements of the one-dimensional algebras $\mathfrak s_{1.1}$ , …, $\mathfrak s_{1.7}$ from Lemma 4 up to neglecting the summands with $\mathcal{I}$ due to the possibility of linearly combining $Q_1$ with $Q_2$ . This results in the subalgebras $\mathfrak s_{2.9}$ , …, $\mathfrak s_{2.15}$ . Hence, below we assume that $(a_{23},a_{22},a_{21},a_{20})\ne (0,0,0,0)$ and, moreover, $\textrm{rank}(a_{ij})^{i=1,2}_{j=0,\dots,3}=2$ if $\hat Q_1=\hat Q_2=0$ .

$\boldsymbol{(\mathcal{P}^t,0).}$ The closedness of $\langle Q_1,Q_2\rangle$ with respect to the Lie bracket implies $a_{23}=a_{22}=a_{21}=a_{13}a_{20}=0$ and thus $a_{13}=0$ since then $a_{20}\ne 0$ . Hence, $Q_1=\mathcal{P}^t+b_1\mathcal{I}$ and $Q_2=\mathcal{P}^0+b_2\mathcal{I}$ .

If $b_2=0$ , we can rescale $b_1$ to $\delta$ by combining $\mathscr D(\epsilon )_*$ with scaling the entire vector field $Q^1$ . This results in the subalgebras $\mathfrak s_{2.2}^\delta$ .

Let $b_2\ne 0$ . Simultaneously acting by $\mathscr D(\epsilon )_*$ with the suitable value of $\epsilon$ , rescaling $b_2$ and, if $b_2\lt 0$ , acting by $\mathscr J_*$ and alternating the sign of $Q^2$ , we can set $b_2=1$ . Then, the pushforward $\mathscr P^1(b_1)_*$ preserves $Q^2$ , and $\mathscr P^1(b_1)_*Q_1=\mathcal{P}^t+b_1 Q^2$ . After linearly combining $Q_1$ with $Q^2$ , we obtain the subalgebras $\mathfrak s_{2.3}$ .

$\boldsymbol{(\mathcal{D},0).}$ In view of Lemma 4, we can reduce $Q_1$ to $\mathcal{D}+\mu \mathcal{I}$ modulo the $G^{\textrm{ess}}$ -equivalence. The condition $[Q_1,Q_2]\in \langle Q_1,Q_2\rangle$ and the commutation relations of $\mathcal{D}$ with basis elements of $\mathfrak r$ imply that up to scaling of $Q^2$ , we have

\begin{equation*} (a_{23},3a_{22},3a_{21},a_{20},b_2)\in \{(1,0,0,0,0),(0,1,0,0,0),(0,0,1,0,0),(0,0,0,1,0)\}, \end{equation*}

where the first and the second values are reduced by $\mathscr K_*^{\prime}$ to the third and the fourth values, respectively. Thus, we obtain the subalgebras $\mathfrak s_{2.4}^\mu$ and $\mathfrak s_{2.5}^\mu$ .

$\boldsymbol{(\mathcal{P}^t}+\boldsymbol{\mathcal{K},0).}$ This pair is not appropriate since, from the closedness of $\langle Q_1,Q_2\rangle$ with respect to the Lie bracket, we straightforwardly derive $a_{2j}=0$ , $j=0,\dots,3$ .

$\boldsymbol{(0,0).}$ Using Lemma 4, from the very beginning we can set $Q_1\in \{\mathcal{P}^2+\varepsilon \mathcal{P}^0,\mathcal{P}^1,\mathcal{P}^0\}$ . Since $\textrm{rank}(a_{ij})^{i=1,2}_{j=0,\dots,3}=2$ here, simultaneously with $b_1=0$ we can set $b_2=0$ using the suitable pushforward $\mathscr P^1(\epsilon )_*$ .

For $Q=a_3\mathcal{P}^3+3a_2\mathcal{P}^2+3a_1\mathcal{P}^1+a_0\mathcal{P}^0$ with $\bar a\,:\!=(a_0,a_1,a_2,a_3)\ne (0,0,0,0)$ , we denote

\begin{equation*} {\textrm {Discr}}(Q)\,:\!=a_0^2a_3^2-6a_0a_1a_2a_3+4a_0a_2^3-3a_1^2a_2^2+4a_1^3a_3. \end{equation*}

The association with cubic binary forms implies that $Q\in \{\mathcal{P}^0,\mathcal{P}^1\}$ modulo the $G^{\textrm{ess}}$ -equivalence if and only if ${\textrm{Discr}}(Q)=0$ , cf. [Reference Olver24, Example 2.22]. To distinguish inequivalent subalgebras in the present classification case, we will consider lines in the space run by $\bar a$ , where ${\textrm{Discr}}(Q)=0$ . We call such lines singular. It is obvious that two-dimensional subalgebras in $\langle \mathcal{P}^0,\mathcal{P}^1,\mathcal{P}^2,\mathcal{P}^3\rangle$ are $G^{\textrm{ess}}$ -inequivalent if they possess different numbers of singular lines associated with their elements or, more precisely, different sets of multiplicities of such singular lines.

If $Q_1=\mathcal{P}^0$ , then the condition $[Q_1,Q_2]\in \langle Q_1,Q_2\rangle$ implies $a_{23}=0$ , and we can set $a_{20}=0$ at any step by linearly combining $Q_2$ with $Q_1$ .

The case $a_{22}=0$ corresponds to the subalgebra $\mathfrak s_{2.6}$ . ${\textrm{Discr}}(Q)=0$ for any $Q\in \mathfrak s_{2.6}$ , that is, the subalgebra $\mathfrak s_{2.6}$ possesses an infinite number of singular lines.

For $a_{22}\ne 0$ , we divide $Q_2$ by $a_{22}$ , act by $\mathscr P^t(\epsilon )_*$ with $\epsilon =a_{21}/(2 a_{22})$ and reset $a_{20}=0$ , linearly combining $Q_2$ with $Q_1$ , which gives the basis of the subalgebra $\mathfrak s_{2.7}$ . For an arbitrary vector field $Q=a_0\mathcal{P}^0+3a_2\mathcal{P}^2$ in $\mathfrak s_{2.7}$ , we obtain ${\textrm{Discr}}(Q)=4a_0a_2^{\,3}$ , and thus, the subalgebra $\mathfrak s_{2.7}$ possesses one singular line of multiplicity one and one singular line of multiplicity three.

If $Q_1=\mathcal{P}^1$ , then the condition $[Q_1,Q_2]\in \langle Q_1,Q_2\rangle$ implies $a_{22}=0$ , and we can set $a_{21}=0$ at any step, linearly combining $Q_2$ with $Q_1$ . We can assume $a_{23}\ne 0$ since otherwise we again obtain the subalgebra $\mathfrak s_{2.6}$ . Rescaling the basis element $Q_2$ , we set $a_{23}=1$ . The subalgebra $\langle \mathcal{P}^1,\mathcal{P}^3\rangle$ is mapped by $\mathscr K_*^{\prime}$ to the subalgebra $\mathfrak s_{2.7}$ . This is why the coefficient $a_{20}$ should be nonzero as well. The action by $\mathscr D(\epsilon )_*$ with the suitable value of $\epsilon$ allows us to set $a_{20}=\varepsilon$ , that is, we derive the subalgebras $\mathfrak s_{2.8}^\varepsilon$ . For an arbitrary vector field $Q=3a_1\mathcal{P}^1+a_3(\mathcal{P}^3+\varepsilon \mathcal{P}^0)$ in $\mathfrak s_{2.8}^\varepsilon$ , we obtain ${\textrm{Discr}}(Q)=a_3(4a_1^3+a_3^3)$ . This means that each of the subalgebras $\mathfrak s_{2.8}^\varepsilon$ possesses two singular line of multiplicity one.

The consideration of the case $Q_1=\mathcal{P}^2+\varepsilon \mathcal{P}^0$ is the most complicated. The span $\langle Q_1,Q_2\rangle$ is closed with respect to the Lie bracket if and only if $a_{21}=-\varepsilon a_{23}$ . The coefficient $a_{22}$ is set to zero by linearly combining $Q_2$ with $Q_1$ . If $a_{23}=0$ , then $a_{21}=0$ as well, and we again obtain the subalgebra $\mathfrak s_{2.7}$ . Let then further $a_{23}\ne 0$ . We scale $Q_2$ to set $a_{23}=1$ , which leads to $Q_2=\mathcal{P}^3-3\varepsilon \mathcal{P}^1+\alpha \mathcal{P}^0$ with $\alpha \,:\!=a_{20}$ . We intend to show that depending on the value of $(\varepsilon,\alpha )$ , the subalgebra $\mathfrak s\,:\!=\langle Q_1,Q_2\rangle$ is $G^{\textrm{ess}}$ -equivalent to either the subalgebra $\mathfrak s_{2.7}$ or the subalgebra $\mathfrak s_{2.8}$ .

We compute that

\begin{equation*} {\textrm {Discr}}(Q)=12\varepsilon a_2^{\,4}+4\alpha a_3a_2^{\,3}+24a_3^{\,2}a_2^{\,2} +12\varepsilon \alpha a_3^{\,3}a_2+a_3^{\,4}\alpha ^2-4a_3^{\,4}\varepsilon \end{equation*}

for a nonzero vector field $Q=3a_2Q_1+a_3Q_2$ in $\mathfrak s$ . If $a_3=0$ and thus $a_2\ne 0$ , then ${\textrm{Discr}}(Q)=12\varepsilon a_2^{\,4}\ne 0$ , that is, the corresponding line is not singular. Hence, we can further set $a_3\ne 0$ and assume without loss of generality that $a_3=1$ . Thus, finding singular lines associated with the subalgebra $\mathfrak s$ reduces to solving the quartic equation ${\textrm{Discr}}(Q)=0$ with respect to $a_2$ . The substitution $a_2=z-\varepsilon \alpha/12$ and the division of the left-hand side by $12\varepsilon$ reduce this equation to its monic depressed counterpart $z^4+pz^2+qz+r=0$ , where

\begin{equation*} p\,:\!=-\frac 1{24}\alpha ^2+2\varepsilon,\quad q\,:\!=\frac \varepsilon {216}\alpha ^3+\frac 23\alpha,\quad r\,:\!=-\frac 1{6912}\alpha ^4+\frac \varepsilon {72}\alpha ^2-\frac 13. \end{equation*}

To find the number of (real) roots of this equation, following [Reference Arnon3, p. 53] we compute

\begin{align*} &\delta \,:\!=256r^3-128p^2r^2+144pq^2r+16p^4r-27q^4-4p^3q^2= -\frac 1{432}(\alpha ^2+16\varepsilon )^4,\\ &L\,:\!=8pr-9q^2-2p^3=-\frac \varepsilon{12}(\alpha ^2+16\varepsilon )^2. \end{align*}

Thus, there are only two cases for the number of roots of the equation ${\textrm{Discr}}(Q)=0$ . Note that in both cases, this number is not zero. Therefore, as indicated above, the subalgebra $\mathfrak s$ contains, modulo the $G^{\textrm{ess}}$ -equivalence, at least one of the vector fields $\mathcal{P}^0$ and $\mathcal{P}^1$ , and thus we can use the considerations of the cases where $Q_1=\mathcal{P}^0$ or $Q_1=\mathcal{P}^1$ . This is why the recognition of the corresponding representatives among subalgebras listed in the lemma’s statement depends only on the root structure of the equation ${\textrm{Discr}}(Q)=0$ .

1. If either $\varepsilon =-1$ and $\alpha \ne \pm 4$ or $\varepsilon =1$ , then $\delta \lt 0$ , and hence the equation ${\textrm{Discr}}(Q)=0$ has two roots of multiplicity one, which corresponds to two singular lines of multiplicity one for the subalgebra $\mathfrak s$ . Therefore, this algebra is $G^{\textrm{ess}}$ -equivalent to one of the subalgebras $\mathfrak s_{2.8}^\varepsilon$ .

2. If $\varepsilon =-1$ and $\alpha =\pm 4$ , then $\delta =L=0$ and $p\lt 0$ , and thus, the equation ${\textrm{Discr}}(Q)=0$ has one root of multiplicity one and one root of multiplicity three, which corresponds to one singular line of multiplicity one and one singular line of multiplicity three for the subalgebra $\mathfrak s$ . Hence, this algebra is $G^{\textrm{ess}}$ -equivalent to the subalgebra $\mathfrak s_{2.7}$ .

We can also classify one and two-dimensional subalgebras of the entire algebra $\mathfrak{g}$ up to the $G$ -equivalence and show that only subalgebras of $\mathfrak{g}^{\textrm{ess}}$ are essential in the course of classifying Lie reductions of the equation (2). Thus, according to the decomposition $G=G^{\textrm{ess}}\ltimes G^{\textrm{lin}}$ (see Section 3), an arbitrary transformation $\Phi$ from $G$ can be represented in the form $\Phi =\mathscr F\circ \mathscr Z(\,f)$ , where $\mathscr F\in G^{\textrm{ess}}$ and $\mathscr Z(\,f)\in G^{\textrm{lin}}$ . To exhaustively describe the adjoint action of $G$ on the algebra $\mathfrak{g}$ , in view of the decomposition $\mathfrak{g}=\mathfrak{g}^{\textrm{ess}}\mathbin{\mbox{${\,\,\rule{.4pt}{1.07ex}}{\!\!\in }$}}\mathfrak{g}^{\textrm{lin}}$ it suffices to supplement the adjoint action of $G^{\textrm{ess}}$ on $\mathfrak{g}^{\textrm{ess}}$ with the adjoint actions of $G^{\textrm{ess}}$ on $\mathfrak{g}^{\textrm{lin}}$ and of $G^{\textrm{lin}}$ on $\mathfrak{g}^{\textrm{ess}}$ , $\mathscr Z(\,f)_*Q=Q+Q[\,f]\partial _u$ and $\mathscr F_*\mathcal{Z}(\,f)=\mathcal{Z}(\mathscr F_*\,f)$ , whereas the adjoint action of $G^{\textrm{lin}}$ on $\mathfrak{g}^{\textrm{lin}}$ is trivial. Here $Q$ is an arbitrary vector field from $\mathfrak{g}^{\textrm{ess}}$ , $\mathcal{Q}[\,f]$ denotes the evaluation of the characteristic $Q[u]$ of $Q$ at $u=\,f$ , and $\mathscr F$ is an arbitrary transformation from $G^{\textrm{ess}}$ .

The classification of subalgebras of $\mathfrak{g}$ is based on the classification of subalgebras of $\mathfrak{g}^{\textrm{ess}}$ . This is due to the fact that subalgebras $\mathfrak s_1$ and $\mathfrak s_2$ of $\mathfrak{g}$ are definitely $G$ -inequivalent if $\pi _{\mathfrak{g}^{\textrm{ess}}}\mathfrak s_1$ and $\pi _{\mathfrak{g}^{\textrm{ess}}}\mathfrak s_2$ are $G^{\textrm{ess}}$ -inequivalent. Here $\pi _{\mathfrak{g}^{\textrm{ess}}}$ denotes the natural projection of $\mathfrak{g}$ onto $\mathfrak{g}^{\textrm{ess}}$ under the decomposition $\mathfrak{g}=\mathfrak{g}^{\textrm{ess}}\dotplus \mathfrak{g}^{\textrm{lin}}$ in the sense of vector spaces. In the course of classifying the subalgebras $\mathfrak s$ of $\mathfrak{g}$ of a fixed (finite) dimension, we partition the set of these subalgebras into subsets such that each subset consists of the subalgebras with the same (up to the $G^{\textrm{ess}}$ -equivalence) projection $\pi _{\mathfrak{g}^{\textrm{ess}}}\mathfrak s$ . As a result, each of these subsets contains no subalgebra being equivalent to a subalgebra from another subset.

The classification of one- and two-dimensional subalgebras of $\mathfrak{g}$ up to the $G$ -equivalence is given in the following two assertions.

Proof. Consider an arbitrary two-dimensional subalgebra $\mathfrak s$ of $\mathfrak{g}$ , and let $Q_1$ and $Q_2$ be its basis elements, $\mathfrak s=\langle Q_1,Q_2\rangle$ with $[Q_1,Q_2]\in \mathfrak s$ . Due to the decomposition $\mathfrak{g}=\mathfrak{g}^{\textrm{ess}}\mathbin{\mbox{${\,\,\rule{.4pt}{1.07ex}}{\!\!\in }$}}\mathfrak{g}^{\textrm{lin}}$ , each $Q_i$ , $i=1,2$ , can be represented in the form $Q_i=\hat Q_i+\mathcal{Z}(\,f^i)$ , where $\hat Q_i\in \mathfrak{g}^{\textrm{ess}}$ and the function $\,f^i$ is a solution of the equation (2). As the principal $G$ -invariant value in the course of classifying two-dimensional subalgebras of $\mathfrak{g}$ , we choose the dimension $d\,:\!=\dim \pi _{\mathfrak{g}^{\textrm{ess}}}\mathfrak s$ of the projection of $\mathfrak s$ onto $\mathfrak{g}^{\textrm{ess}}$ .

$\boldsymbol{d=2.}$ Up to the $G^{\textrm{ess}}$ -equivalence, the pair $(\hat Q_1,\hat Q_2)$ can be assumed to be the chosen basis of one of the subalgebras $\mathfrak s^{\mu }_{2.1}$ , …, $\mathfrak s_{2.15}$ of $\mathfrak{g}^{\textrm{ess}}$ listed in Lemma 5.

For the subalgebras $\mathfrak s_{2.9}$ , …, $\mathfrak s_{2.15}$ , we have $Q_2=\mathcal{I}$ . Pushing $\mathfrak s$ forward with $\mathscr Z(h)$ , where $h=-\,f^2$ , we can set $\,f^2=0$ . Then, the commutation relation $[Q_1,Q_2]=0$ implies $\,f^1=0$ .

For the subalgebras $\mathfrak s_{2.1}^\mu$ , …, $\mathfrak s_{2.8}^\varepsilon$ , let $\kappa _1$ and $\kappa _2$ be the constants such that $[\hat Q_1,\hat Q_2]=\kappa _1\hat Q_1+\kappa _2\hat Q_2$ . Then, the functions $\,f^1$ and $\,f^2$ satisfy the constraint $\hat Q_1[\,f^2]-\hat Q_2[\,f^1]=\kappa _1\,f^1+\kappa _2\,f^2$ . Hence, we can set them to be identically zero using the pushforward $\mathscr Z(h)_*$ , where $h$ is a solution of the overdetermined system

\begin{equation*}h_t+xh_y=h_{xx},\quad \hat Q_1[h]+\,f^1=0,\quad \hat Q_2[h]+\,f^2=0\end{equation*}

with respect to $h$ .Footnote ⁵

In total, this results in the first family of subalgebras from the lemma’s statement.

$\boldsymbol{d=1.}$ Without loss of generality, we can assume $\hat Q_2=0$ up to linearly recombining the basis elements $Q_1$ and $Q_2$ and thus $\,f^2\ne 0$ , whereas modulo the $G^{\textrm{ess}}$ -equivalence, the vector field $\hat Q_1$ can be assumed to be the basis element of one of the subalgebras $\mathfrak s_{1.1}$ , …, $\mathfrak s_{1.8}$ of $\mathfrak{g}^{\textrm{ess}}$ that are listed in Lemma 4. Then Lemma 4 also implies that $\,f^1=0$ up to the $G^{\textrm{ess}}$ -equivalence.

For the subalgebras $\mathfrak s_{1.1}$ , …, $\mathfrak s_{1.7}$ , we have $\hat Q_1\notin \langle \mathcal{I}\rangle$ . Following the proof of Lemma 6, we can set the function $\,f^1$ to be identically zero. Then, the closedness of the subalgebra $\mathfrak s$ with respect to the Lie bracket implies the constraint $\hat Q_1[\,f^2]=\lambda \,f^2$ . In other words, the function $\,f^2$ is a $\langle \hat Q_1+\lambda \mathcal{I}\rangle$ -invariant solution of the equation (2). Transformations from the group $\textrm{St}_{G^{\textrm{ess}}}(\langle \hat Q_1\rangle )$ allow us to set, depending on the value of $\hat Q_1$ , the restrictions on $\lambda$ that are presented in item 2 of the lemma’s statement. The transformations from $\textrm{St}_{G^{\textrm{ess}}}(\langle \hat Q_1\rangle )$ that preserve $\lambda$ also induce an equivalence relation on the corresponding set run by $\,f^2$ .

Since $\hat Q_1=\mathcal{I}$ for the subalgebra $\mathfrak s_{1.8}$ , in the way described in the proof of Lemma 6, we can set the function $\,f^1$ to be identically zero. Then the only commutation relation of the algebra $\mathfrak s$ is $[Q_1,Q_2]=Q_2$ , which implies no constraint for the function $\,f^2$ , and $\textrm{St}_{G^{\textrm{ess}}}(\langle \mathcal{I}\rangle )=G^{\textrm{ess}}$ . Hence in this case, the parameter function $\,f^2$ should run through a fixed complete set of $G^{\textrm{ess}}$ -inequivalent nonzero solutions of the equation (2).

Thus, we obtained the second and third families of subalgebras listed in the lemma’s statement.

$\boldsymbol{d = 0.}$ This case is obvious since then $\hat Q_1=\hat Q_2=0$ . In other words, the subalgebra $\langle Q_1,Q_2\rangle$ is of the form $\langle \mathcal{Z}(\,f^1),\mathcal{Z}(\,f^2)\rangle$ , where $\,f^1$ and $\,f^2$ are linearly independent solutions of the equation (2), which should be chosen modulo the $G^{\textrm{ess}}$ -equivalence and linearly recombining.

5. Lie reductions of codimension one

The classification of Lie reductions of the equation (2) to partial differential equations with two independent variables is based on the classification of one-dimensional subalgebras of the algebra $\mathfrak{g}^{\textrm{ess}}$ , which is given in Lemma 4. In Table 1, for each of the subalgebras listed therein, we present an associated ansatz for $u$ and the corresponding reduced equation. Throughout this section, the subscripts 1 and 2 of functions depending on $(z_1,z_2)$ denote derivatives with respect to $z_1$ and $z_2$ , respectively. In this and the next sections, we mark Lie reductions and all related objects with complex labels $d.i^*$ , where $d$ , $i$ and $*$ are the dimension of the corresponding subalgebra, its number in the list of $d$ -dimensional subalgebras of the algebra $\mathfrak{g}^{\textrm{ess}}$ , which is given in Lemma 4 for $d=1$ or in Lemma 5 for $d=2$ , and the list of subalgebra parameters for a subalgebra family, respectively. We omit the superscript in the label when it is not essential.

Table 1. $G$ -inequivalent Lie reductions of codimension one

Here $\varepsilon '=\mathop{\textrm{sgn}}\nolimits t$ , $\varepsilon \in \{-1,1\}$ , $\delta \in \{-1,0,1\}$ , $w=w(z_1,z_2)$ is the new unknown function of the new independent variables $(z_1,z_2)$ , and

\begin{equation*} \psi (t,x,y,\mu )=\dfrac {3t^3y^2+t(2x(t^2+1)-3ty)^2}{4(t^2+1)^3}+\mu \arctan t. \end{equation*}

Each of the reduced equations $1.i$ presented in Table 1, $i\in \{1,2^\delta,3^\nu,4^\mu,5^\varepsilon,6,7\}$ , is a linear homogeneous partial differential equation in two independent variables. Hence, its maximal Lie invariance algebra $\mathfrak{g}_{1.i}$ contains the infinity-dimensional abelian ideal $\{h(z_1,z_2)\partial _w\}$ , which is associated with the linear superposition of solutions of reduced equation $1.i$ and thus assumed to be a trivial part of $\mathfrak{g}_{1.i}$ . Here the parameter function $h=h(z_1,z_2)$ runs through the solution set of reduced equation $1.i$ . Moreover, the algebra $\mathfrak{g}_{1.i}$ is the semidirect sum of the above ideal and a finite-dimensional subalgebra $\mathfrak a_i$ called the essential Lie invariance algebra of reduced equation $1.i$ , cf. Section 2. The algebras $\mathfrak a_i$ are the following:

\begin{align*} &\mathfrak a_1=\mathfrak a_3=\mathfrak a_4=\langle w\partial _w\rangle,\\ &\mathfrak a_2^0=\langle \partial _{z_1},\,3z_1\partial _{z_1}\!+z_2\partial _{z_2},\,9z_1^2\partial _{z_1}\!+6z_1z_2\partial _{z_2}\!-(6z_1+z_2^3)w\partial _w,\,w\partial _w\rangle,\\ &\mathfrak a_2^\delta =\langle \partial _{z_1},\,w\partial _w\rangle \quad \mbox{if}\quad \delta \ne 0,\\ &\mathfrak a_5=\mathfrak a_6=\mathfrak a_7=\langle \partial _{z_1},\,\partial _{z_2},\,2z_1\partial _{z_1}\!+z_2\partial _{z_2},\,2z_1\partial _{z_2}\!-z_2w\partial _w,\\ &\phantom{\mathfrak a_5=\mathfrak a_6=\mathfrak a_7=\langle } 4z_1^2\partial _{z_1}\!+4z_1z_2\partial _{z_2}\!-(2z_1+z_2^2)w\partial _w,\,w\partial _w\rangle. \end{align*}

The equation (2) admits hidden symmetries with respect to a Lie reduction if the corresponding reduced equation possesses Lie symmetries that are not induced by Lie symmetries of the original equation (2). To check which Lie symmetries of reduced equation $1.i$ are induced by Lie symmetries of (2), we compute the normaliser of the subalgebra $\mathfrak s_{1.i}$ in the algebra $\mathfrak{g}^{\textrm{ess}}$ :

\begin{align*} &\textrm{N}_{\mathfrak{g}^{\textrm{ess}}}(\mathfrak s_{1.1})=\langle \mathcal{P}^t+\mathcal{P}^3, \mathcal{I}\rangle,\ \; \textrm{N}_{\mathfrak{g}^{\textrm{ess}}}(\mathfrak s_{1.2}^0)=\langle \mathcal{P}^t,\mathcal{D},\mathcal{P}^0,\mathcal{I}\rangle,\ \;\textrm{N}_{\mathfrak{g}^{\textrm{ess}}}(\mathfrak s_{1.2}^\delta )=\langle \mathcal{P}^t,\mathcal{P}^0,\mathcal{I}\rangle \,\ \mbox{if}\,\ \delta \ne 0,\\ &\textrm{N}_{\mathfrak{g}^{\textrm{ess}}}(\mathfrak s_{1.3}^\nu )=\langle \mathcal{D},\mathcal{I}\rangle,\ \; \textrm{N}_{\mathfrak{g}^{\textrm{ess}}}(\mathfrak s_{1.4}^\mu )=\langle \mathcal{P}^t+\mathcal{K},\mathcal{I}\rangle,\ \; \textrm{N}_{\mathfrak{g}^{\textrm{ess}}}(\mathfrak s_{1.5}^\varepsilon )=\langle \mathcal{P}^2,\mathcal{P}^0,\mathcal{P}^3-3\varepsilon \mathcal{P}^1,\mathcal{I}\rangle,\\ &\textrm{N}_{\mathfrak{g}^{\textrm{ess}}}(\mathfrak s_{1.6})=\langle \mathcal{D},\mathcal{P}^3,\mathcal{P}^1,\mathcal{P}^0,\mathcal{I}\rangle,\ \; \textrm{N}_{\mathfrak{g}^{\textrm{ess}}}(\mathfrak s_{1.7})=\langle \mathcal{P}^t,\mathcal{D},\mathcal{P}^2,\mathcal{P}^1,\mathcal{P}^0,\mathcal{I}\rangle. \end{align*}

The algebra of induced Lie symmetries of reduced equation $1.i$ is isomorphic to the quotient algebra $\textrm{N}_{\mathfrak{g}^{\textrm{ess}}}(\mathfrak s_{1.i})/\mathfrak s_{1.i}$ . Thus, all Lie symmetries of the reduced equation $1.i$ are induced by Lie symmetries of the original equation (2) if and only if $\dim \mathfrak a_i=\dim \textrm{N}_{\mathfrak{g}^{\textrm{ess}}}(\mathfrak s_{1.i})-1$ . Comparing the dimensions of $\textrm{N}_{\mathfrak{g}^{\textrm{ess}}}(\mathfrak s_{1.i})$ and $\mathfrak a_i$ , we conclude that all Lie symmetries of reduced equations $1.1$ , $1.3$ and $1.4$ are induced by Lie symmetries of (2). The subalgebras of induced symmetries in the algebras $\mathfrak a_2^\delta$ and $\mathfrak a_5$ , …, $\mathfrak a_7$ are

\begin{align*} &\tilde{\mathfrak a}_2^0=\langle \partial _{z_1},\,3z_1\partial _{z_1}\!+z_2\partial _{z_2},\,w\partial _w\rangle,\quad \tilde{\mathfrak a}_2^\delta =\langle \partial _{z_1},\,w\partial _w\rangle \ \ \mbox{if}\ \ \delta \ne 0,\ \\ &\tilde{\mathfrak a}_5=\langle \partial _{z_2},\,2z_1\partial _{z_2}\!-z_2w\partial _w,\,w\partial _w\rangle,\\ &\tilde{\mathfrak a}_6=\langle \partial _{z_2},\,2z_1\partial _{z_1}\!+z_2\partial _{z_2},\,2z_1\partial _{z_2}\!-z_2w\partial _w,\,w\partial _w\rangle,\\ &\tilde{\mathfrak a}_7=\langle \partial _{z_1},\,\partial _{z_2},\,2z_1\partial _{z_1}\!+z_2\partial _{z_2},\,2z_1\partial _{z_2}\!-z_2w\partial _w,\,w\partial _w\rangle. \end{align*}

For each $i\in \{2^0,5,6,7\}$ , the elements of the complement $\mathfrak a_i/\tilde{\mathfrak a}_i$ are nontrivial hidden symmetries of the equation (2) that are associated with reduction $1.i$ . See Appendix B on an optimised procedure for constructing exact solutions using hidden symmetries.

We discuss reduced equations 1.1–1.7 and present solutions of (2) that can be found using the known solutions of the reduced equations.

Using point transformations, we can further map reduced equations 1.1–1.4 to (1 + 1)-dimensional linear heat equations with potentials, which are of the general form $w_1=w_{22}+Vw$ , where $V$ is an arbitrary function of $(z_1,z_2)$ . These transformations and mapped equations are

1. 1.1. $\tilde z_1=\frac 94\tilde{\varepsilon}z_1,\ \ \tilde z_2=|z_2|^{\frac 32},\ \ \tilde w=|z_2|^{\frac 14}w\ \ {} $ with $\ \ \tilde{\varepsilon}:=\mathop{\textrm{sgn}} z_2$ : $\quad\tilde w_1=\tilde w_{22}-\frac 19\left (\frac{16}{3}\tilde{\varepsilon}\tilde z_1\tilde z_2^{-\frac 23}-\frac 54\tilde z_2^{-2}\right )\tilde w$ ;

2. 1.2 ${^\delta \!.}\phantom{.}\,$ $\tilde z_1=\frac 94 \tilde{\varepsilon} z_1,\ \ \tilde z_2=|z_2|^{\frac 32},\ \ \tilde w=|z_2|^{\frac 14}w\ \ {} $ with $\ \ \tilde{\varepsilon}:=\mathop{\textrm{sgn}}z_2$ : $\quad\tilde w_1=\tilde w_{22}-\frac 19\left (4\delta \tilde z_2^{-\frac 23}-\frac 54\tilde z_2^{-2}\right )\tilde w$ ;

3. 1.3 $\displaystyle ^\nu \!.$ $\tilde z_1=\frac 94\tilde \varepsilon z_1,\ \ \tilde z_2=|z_2-\frac 32\varepsilon 'z_1|^{\frac 32},\ \ \tilde w=|2z_2-3\varepsilon 'z_1|^{\frac 14}\textrm{e}^{\frac 18z_2(4\varepsilon'z_2-9z_1)}w\ {}$ with $\ \tilde \varepsilon \,:\!=\mathop{\textrm{sgn}}\nolimits\!(z_2-\frac 32\varepsilon 'z_1)$ : $\tilde w_1=\tilde w_{22}+\frac 1{144}\tilde z_2^{-\frac 23}\left( 16\tilde z_1^2-40(\tilde z_2^{\frac 23}+\frac 23\varepsilon ' \tilde z_1)^2+20\tilde z_2^{-\frac 43}-32\varepsilon '\nu \right)\tilde w$ ;

4. 1.4 $\displaystyle ^\mu \!.$ $\tilde z_1=\frac 94\tilde \varepsilon z_1,\ \ \tilde z_2=|z_2|^{\frac 32},\ \ \tilde w=|z_2|^{\frac 14}\textrm{e}^{\frac 32z_1z_2}w\ \ {}$ with $\ \ \tilde \varepsilon \,:\!=\mathop{\textrm{sgn}}\nolimits z_2$ : $\tilde w_1=\tilde w_{22}+\frac 19\left (\frac 54\tilde z_2^{-2}+10\tilde z_2^{\frac 23}+(4\mu - 9\tilde z_1^2)\tilde z_2^{-\frac 23}\right )\tilde w$ .

The essential Lie invariance algebras of reduced equations 1.1, 1.3 and 1.4 are trivial, and thus, the further consideration of these equations within the classical framework gives no results.

Mapped reduced equation 1.2 $^0$ coincides with the linear heat equation with potential $V=\mu z_2^{-2}$ , where $\mu =\frac 5{36}$ . Hence, we construct the following family of solutions of the equation (2):

\begin{align*} \bullet \quad u=|x|^{-\frac 14}\theta ^\mu \left(\frac 94 \tilde{\varepsilon} y,|x|^{\frac 32}\right)\ \ \mbox{with}\ \ \mu =\frac 5{36}, \end{align*}

where $\theta ^\mu =\theta ^\mu (z_1,z_2)$ is an arbitrary solution of the equation $\theta ^\mu _1=\theta ^\mu _{22}+\mu z_2^{-2}\theta ^\mu$ , and $\tilde{\varepsilon}:=\mathop{\textrm{sgn}}x$ . A complete collection of inequivalent Lie invariant solutions of equations of such form with general $\mu \ne 0$ is presented in Appendix A.

The essential Lie invariance algebra of reduced equation 1.2 $^\delta$ with $\delta \ne 0$ is induced by the subalgebra $\langle \mathcal{P}^0,\mathcal{I}\rangle$ of $\mathfrak{g}^{\textrm{ess}}$ . The subalgebras of $\mathfrak a_2^\delta$ that are appropriate for Lie reduction of reduced equation 1.2 $^\delta$ are exhausted by the subalgebras of the form $\langle \partial _{z_1}\!+\nu w\partial _w\rangle$ , which are respectively induced by the subalgebras $\langle \mathcal{P}^0+\nu \mathcal{I}\rangle$ of $\mathfrak{g}^{\textrm{ess}}$ . This is why the family of solutions of reduced equation 1.2 $^\delta$ that are invariant with respect to $\langle \partial _{z_1}\!+\nu w\partial _w\rangle$ gives a family of solutions of the equation (2) that is contained, up to the $G^{\textrm{ess}}$ -equivalence, in the family of $\langle \mathcal{P}^0\rangle$ -invariant solutions, see reduction 1.7.

Each of reduced equations 1.5–1.7 itself coincides with the classical (1 + 1)-dimensional linear heat equation, which leads to the following families of solutions of the equation (2):

\begin{align*} &\bullet \quad u=|t|^{-\frac 12}\textrm{e}^{-\frac{x^2}{4t}}\theta (z_1,z_2),\ \ \mbox{where}\ \ z_1=\frac 13{t^3}+2\varepsilon t-t^{-1},\ \ z_2=2y-(t+\varepsilon t^{-1})x,\\ &\bullet \quad u=\theta (z_1,z_2),\ \ \mbox{where}\ \ z_1=\frac 13t^3,\ \ z_2=y-tx,\\ &\bullet \quad u=\theta (z_1,z_2),\ \ \mbox{where}\ \ z_1=t,\ \ z_2=x. \end{align*}

Here $\theta =\theta (z_1,z_2)$ is an arbitrary solution of the (1 + 1)-dimensional linear heat equation, $\theta _1=\theta _{22}$ . An enhanced complete collection of inequivalent Lie invariant solutions of this equation was presented in [Reference Vaneeva, Popovych and Sophocleous40, Section A], following Examples 3.3 and 3.17 in [Reference Olver25]. These solutions exhaust, up to combining the $G^{\textrm{ess}}$ -equivalence and linear superposition with each other, the set of known exact solutions of this equation that are expressed in a closed form in terms of elementary and special functions.

6. Lie reductions of codimension two

Based on the list of $G^{\textrm{ess}}$ -inequivalent two-dimensional subalgebras from Lemma 5, we carry out the exhaustive classification of codimension-two Lie reductions of the equation (2). In Table 2, for each of these subalgebras, we present an associated ansatz for $u$ and the corresponding reduced ordinary differential equation.

Table 2. $G$ -inequivalent Lie reductions of codimension two

Here $\varepsilon '=\mathop{\textrm{sgn}}\nolimits t$ , $\varphi =\varphi (\omega )$ is the new unknown function of the invariant independent variable $\omega$ .

We find (nonzero) exact solutions of the constructed reduced equations and present the associated exact solutions of the equation (2). Hereafter, $C_0$ , $C_1$ and $C_2$ are arbitrary constants.

2.1. Only this codimension-two Lie reduction results in new exact solutions of (2) in comparison with those constructed using Lie reductions with respect to one-dimensional subalgebras. The substitution $\varphi =|\omega |^{\frac 13}\textrm{e}^{-\frac{\omega }9}\tilde \varphi (\omega )$ maps reduced equation 2.1 to a Kummer’s equation

\begin{align*} 9\omega \tilde \varphi _{\omega \omega }+(12-\omega )\tilde{\varphi }_{\omega }=\frac{\mu +1}3\tilde \varphi, \end{align*}

whose general solution is

\begin{equation*} \tilde \varphi =C_1M\left (\frac{\mu+1}3,\frac 43,\frac{\omega}9 \right )+C_2 U\left (\frac {\mu+1}3,\frac 43,\frac{\omega}9\right ), \end{equation*}

where $M(a,b,z)$ and $U(a,b,z)$ denote the standard solutions of general Kummer’s equation $z\varphi _{zz}+(b-z)\varphi _z-a\varphi =0$ . After taking $\kappa \,:\!=\frac 13(\mu +1)$ instead of $\mu$ as an arbitrary constant parameter, the corresponding family of particular solutions of the equation (2) can be represented in the form

\begin{align*} \bullet \quad u=x|y|^{\kappa -\frac 43}\textrm{e}^{-\tilde \omega } \left (C_1M\left (\kappa,\frac 43,\tilde \omega \right )+C_2U\left (\kappa,\frac 43,\tilde \omega \right )\right )\quad \mbox{with}\quad \tilde \omega \,:\!=\frac{x^3}{9y}. \end{align*}

It also admits the equivalent representation

\begin{align*} \bullet \displaystyle\quad u=x^{-1}|y|^{\kappa -\frac 23}\textrm{e}^{-\frac{\tilde \omega }2} \left (C_1M_{\frac 23-\kappa,\frac 16}(\tilde \omega )+C_2W_{\frac 23-\kappa,\frac 16}(\tilde \omega )\right )\quad \mbox{with}\quad \tilde \omega \,:\!=\frac{x^3}{9y} \end{align*}

in terms of the Whittaker functions $M_{a,b}(z)$ and $W_{a,b}(z)$ , which constitute the fundamental solution set of the Whittaker equation (16), due to a relation of these functions to Kummer’s functions,

\begin{align*} \displaystyle & M_{a,b}(z)\,:\!=\textrm{e}^{-\frac z2}z^{b+\frac 12}M\left (b-a+\frac 12,1+2b,z\right ),\\ & W_{a,b}(z)\,:\!=\textrm{e}^{-\frac z2}z^{b+\frac 12}U\left (b-a+\frac 12,1+2b,z\right ). \end{align*}

The rest of the reductions only lead to solutions of the equation (2), each of which is $G^{\textrm{ess}}$ -equivalent to a solution of form 1.6 or 1.7 given in Section 5. Indeed, each of the corresponding subalgebras $\mathfrak s_{2.2}$ , …, $\mathfrak s_{2.8}$ contains, up to the $G^{\textrm{ess}}$ -equivalence for the subalgebra $\mathfrak s_{2.3}$ , at least one of the vector fields $\mathcal{P}_0$ and $\mathcal{P}_1$ . Nevertheless, we discuss these reductions below in order to explicitly present solutions of (2) that are invariant with respect to two-dimensional algebras of Lie-symmetry vector fields of (2).

2.2 $^\delta$ . Reduced equation 2.2 $^\delta$ trivially integrates to

\begin{align*} \varphi = \begin{cases} C_1\textrm{e}^\omega +C_2\textrm{e}^{-\omega }\quad &\mbox{if}\ \ \delta =1,\\ C_1\omega +C_2\quad &\mbox{if}\ \ \delta =0,\\ C_1\sin \omega +C_2\cos \omega \quad &\mbox{if}\ \ \delta =-1. \end{cases} \end{align*}

2.3. Reduced equation 2.3 is the Airy equation, whose general solution is $\varphi =C_1\textrm{Ai}(\omega )+C_2\textrm{Bi}(\omega )$ .

2.4, 2.5. Substituting $\varphi =\omega \textrm{e}^{-\frac 34\varepsilon^\prime\omega ^2}\tilde \varphi (\omega )$ and $\varphi=\omega\textrm{e}^{-\frac{1}{4}\varepsilon^\prime\omega^2}\tilde{\varphi}(\omega)$ into reduced equations 2.4 and 2.5, respectively, we obtain the equations

\begin{align*} &2\omega \tilde \varphi _{\omega \omega }+(4-3\varepsilon^\prime\omega ^2)\tilde \varphi _\omega =\varepsilon '(\mu +4)\omega \tilde \varphi,\\ &2\omega \tilde \varphi _{\omega \omega }+(4-\varepsilon '\omega ^2)\tilde \varphi _{\omega }=\varepsilon '\mu \omega \tilde \varphi, \end{align*}

whose general solutions are

\begin{align*} &\tilde \varphi =C_1M\left (\frac \mu 6+\frac 23,\frac 32,\frac 34\varepsilon '\omega ^2\right ) +C_2 U\left (\frac \mu 6+\frac 23,\frac 32,\frac 34\varepsilon '\omega ^2\right ), \\[1ex] &\tilde \varphi =C_1M\left (\frac \mu 2,\frac 32,\frac 14\varepsilon '\omega ^2\right ) +C_2 U\left (\frac \mu 2,\frac 32,\frac 14\varepsilon '\omega ^2\right ). \end{align*}

2.6–2.8. Here the reduced equations are ordinary first-order differential equations, which are trivially integrated to

\begin{align*}{2.6.}\ \varphi =C_0,\quad{2.7.}\ \varphi =\frac{C_0}{\sqrt{|\omega|} }\,,\quad{2.8.}\ \varphi =\frac{C_0}{\sqrt{|2\omega ^3-\varepsilon | }}\,. \end{align*}

Substituting the above general solutions of reduced equations 2.2–2.8 into the corresponding ansatzes, we construct the following families of particular solutions of the equation (2), which are contained (as mentioned earlier, up to the $G^{\textrm{ess}}$ -equivalence for the fourth family) in the solution families obtained in Section 5 using Lie reductions 1.6 and 1.7:

\begin{align*} &\bullet \quad u=C_1\textrm{e}^{t+x}+C_2\textrm{e}^{t-x},\qquad \qquad \bullet \quad u=C_1x+C_2,\qquad \qquad \bullet \quad u=C_1\textrm{e}^{-t}\sin x+C_2\textrm{e}^{-t}\cos x, \\ &\bullet \quad u=C_1\textrm{e}^y\textrm{Ai}(x)+C_2\textrm{e}^y\textrm{Bi}(x), \end{align*}

\begin{align*} &\bullet \quad u=|t|^{\frac{\mu -5}2}(y-tx)\textrm{e}^{-\tilde \omega } \left (C_1M\left (\frac \mu 6\!+\!\frac 23,\frac 32,\tilde \omega \right )+ C_2U\left (\frac \mu 6\!+\!\frac 23,\frac 32,\tilde \omega \right )\right ) \ \mbox{with}\ \,\tilde \omega \,:\!=\frac{3(y-tx)^2}{4t^3}, \\ &\bullet \quad u=|t|^{\frac{\mu -3}2}x\textrm{e}^{-\tilde \omega } \left (C_1M\left (\frac \mu 2,\frac 32,\tilde \omega \right )+ C_2U\left (\frac \mu 2,\frac 32,\tilde \omega \right )\right ) \ \ \mbox{with}\ \ \tilde \omega \,:\!=\frac{x^2}{4t}, \\ &\bullet \quad u=C_0,\qquad \qquad \bullet \quad u=\frac{C_0}{\sqrt{|t|}}\textrm{e}^{-\frac{x^2}{4t}},\qquad \qquad \bullet \quad u=\frac{C_0}{\sqrt{|2t^3-\varepsilon |}}\textrm{e}^{-\frac 32\frac{(y-tx)^2}{2t^3-\varepsilon }}. \end{align*}

Considering nonzero solutions from this section up to the $G^{\textrm{ess}}$ -equivalence, we can set the constant $C_0$ and one of the constants $C_1$ and $C_2$ to one. Taking into account the linear superposition principle for solutions of the equation (2), it suffices to present only linearly independent solutions.

7. Lie reductions to algebraic equations

As for an equation in three independent variables, the maximal codimension of Lie reductions for the equation (2) is equal to three, and such reductions lead to algebraic equations. Let us show that these reductions do not give new explicit solutions of the equation (2).

Any three-dimensional subalgebra of the algebra $\mathfrak{g}$ that is appropriate for Lie reduction is $G$ -equivalent to a subalgebra $\mathfrak s$ of $\mathfrak{g}^{\textrm{ess}}$ . If the subalgebra $\mathfrak s$ nontrivially intersects the radical $\mathfrak r$ of $\mathfrak{g}^{\textrm{ess}}$ , then any $\mathfrak s$ -invariant solution of (2) is $G^{\textrm{ess}}$ -equivalent to a solution from one of families 1.5–1.7 constructed in Section 5. Otherwise, the natural projection of the subalgebra $\mathfrak s$ onto the Levi factor $\mathfrak f=\langle \mathcal{P}^t,\mathcal{D},\mathcal{K}\rangle$ under the Levi decomposition $\mathfrak{g}^{\textrm{ess}}=\mathfrak f\mathbin{\mbox{${\,\,\rule{.4pt}{1.07ex}}{\!\!\in }$}}\mathfrak r$ is three-dimensional. Therefore, the subalgebra $\mathfrak s$ is nonsolvable, and thus it is simple, that is, it is a Levi factor of the algebra $\mathfrak{g}^{\textrm{ess}}$ . According to the Levi–Malcev theorem, stating that any two Levi factors of a finite-dimensional Lie algebra are conjugate by an inner automorphism generated by elements of the corresponding nilradical, the subalgebra $\mathfrak s$ is $G^{\textrm{ess}}$ -equivalent to $\mathfrak f$ . An ansatz constructed using the subalgebra $\mathfrak f$

\begin{align*} \bullet \quad u=C_0y^{-\frac 23}\textrm{e}^{-\frac{x^3}{9y}}, \end{align*}

where $C_0$ plays the role of the unknown constant, identically satisfies the equation (2). Up to the $G^{\textrm{ess}}$ -equivalence, we can set the constant $C_0$ equal to one. This solution belongs to the family of particular solutions obtained with reduction 2.1 $^\mu$ in Section 6, where $\mu =0$ or, equivalently, $\kappa =1/3$ , since $U\!\left (\frac 13,\frac 43,z\right )=z^{-\frac 13}$ .

8. Generalised reductions and generating solutions with symmetry operators

We can construct more general families of exact solutions of the equation (2) using generalised symmetries. Since the equation (2) is linear and homogeneous, each first-order linear differential operator in total derivatives that is associated with an essential Lie symmetry of this equation is its recursion operator. Thus, it is obvious that the equation (2) admits, in particular, the generalised vector fields $ ((\mathfrak P^2+\varepsilon \mathfrak P^0)^nu )\partial _u$ , $ ((\mathfrak P^1)^nu )\partial _u$ and $ ((\mathfrak P^0)^nu )\partial _u$ as its generalised symmetries. Here and in what follows,

\begin{align*} \mathfrak P^3\,:\!=3t^2\textrm{D}_x+t^3\textrm{D}_y-3(y-tx),\ \ \mathfrak P^2\,:\!=2t\textrm{D}_x+t^2\textrm{D}_y+x,\ \ \mathfrak P^1\,:\!=\textrm{D}_x+t\textrm{D}_y,\ \ \mathfrak P^0\,:\!=\textrm{D}_y \end{align*}

are the differential operators in total derivatives that are associated with the Lie-symmetry vector fields $-\mathcal{P}^3$ , $-\mathcal{P}^2$ , $-\mathcal{P}^1$ and $-\mathcal{P}^0$ of the equation (2), and $\textrm{D}_x$ and $\textrm{D}_y$ denote the operators of total derivatives with respect to $x$ and $y$ , respectively. The corresponding invariant solutions are constructed by means of solving the equation (2) with the associated invariant surface condition, $(\mathfrak P^2+\varepsilon \mathfrak P^0)^nu=0$ , $(\mathfrak P^1)^nu=0$ or $(\mathfrak P^0)^nu=0$ , respectively. For $n=2$ , it is easy to derive the representation for such solutions in terms of solutions of the (1 + 1)-dimensional linear heat equation,

\begin{align*} & \bullet \quad u=|t|^{-\frac 12}\textrm{e}^{-\frac{x^2}{4t}}\left ( (t-\varepsilon t^{-1})\theta ^1_2-\frac x{2t}\theta ^1+\theta ^0 \right ),\end{align*}

\begin{align*}&\quad\quad \mbox{where}\ \ z_1=\frac 13t^3+2\varepsilon t-t^{-1},\ \ z_2=2y-(t+\varepsilon t^{-1})x, \\[1ex] &\bullet \quad u=x\theta ^1-t^2\theta ^1_2+\theta ^0,\ \ \mbox{where}\ \ z_1=\frac 13t^3,\ \ z_2=y-tx, \\[1ex] &\bullet \quad u=4y\theta ^1_{22}+x^2\theta ^1_2-x\theta ^1+\theta ^0,\ \ \mbox{where}\ \ z_1=t,\ \ z_2=x. \end{align*}

Here $\theta ^i=\theta ^i(z_1,z_2)$ is an arbitrary solution of the (1 + 1)-dimensional linear heat equation, $\theta ^i_1=\theta ^i_{22}$ , $i=0,1$ . As in Section 5, the subscripts 1 and 2 of functions depending on $(z_1,z_2)$ denote derivatives with respect to $z_1$ and $z_2$ , respectively.

In fact, for each of the above generalised symmetries, we can construct the corresponding family of invariant solutions using algebraic tools and the structure of the algebra $\mathfrak{g}$ , or, more specifically, of its radical $\mathfrak r$ . Since $[\mathcal{P}^2+\varepsilon \mathcal{P}^0,\mathcal{P}^1]=\mathcal{I}$ , $[\mathcal{P}^1,\mathcal{P}^2]=-\mathcal{I}$ and $[\mathcal{P}^0,\mathcal{P}^3]= 3\mathcal{I}$ , then

(9) \begin{align} \begin{split}& (\mathfrak P^2+\varepsilon \mathfrak P^0)^n(\mathfrak P^1)^{n-1}\left(|t|^{-\frac 12}\textrm{e}^{-\frac{x^2}{4t}}\theta (z_1,z_2)\right)=0,\\[1ex]& (\mathfrak P^1)^n(\mathfrak P^2)^{n-1}\left (\theta (z_1,z_2)\right )=0,\\[1ex]& (\mathfrak P^0)^n(\mathfrak P^3)^{n-1}\left (\theta (z_1,z_2)\right )=0 \end{split} \end{align}

for arbitrary solutions $\theta =\theta (z_1,z_2)$ of the (1 + 1)-dimensional linear heat equation, $\theta _1=\theta _{22}$ , whereas the expression in the left-hand side of each equality in (9) in general becomes nonvanishing after replacing the $n$ th power of the first operator by its $(n-1)$ th power. The expressions for the variables $z_1$ and $z_2$ should be taken from the corresponding item in the above list. Recall that the operators $\mathfrak P^0$ , …, $\mathfrak P^3$ preserve the solution set of the equation (2). Therefore, the families of its solutions that are invariant with respect to its generalised symmetries $ ((\mathfrak P^2+\varepsilon \mathfrak P^0)^nu )\partial _u$ , $ ((\mathfrak P^1)^nu )\partial _u$ or $ ((\mathfrak P^0)^nu )\partial _u$ , respectively, take the form

\begin{align*} &\bullet \quad u=\sum _{k=0}^{n-1}(\mathfrak P^1)^k\left (|t|^{-\frac 12}\textrm{e}^{-\frac{x^2}{4t}}\theta ^k(z_1,z_2)\right ) \quad \mbox{with}\quad z_1=\frac 13t^3+2\varepsilon t-t^{-1},\ \ z_2=2y-(t+\varepsilon t^{-1})x, \\ &\bullet \quad u=\sum _{k=0}^{n-1}(\mathfrak P^2)^k\theta ^k(z_1,z_2) \quad \mbox{with}\quad z_1=\frac 13t^3,\ \ z_2=y-tx, \\ &\bullet \quad u=\sum _{k=0}^{n-1}(\mathfrak P^3)^k\theta ^k(z_1,z_2) \quad \mbox{with}\quad z_1=t,\ \ z_2=x, \end{align*}

where $\theta ^k=\theta ^k(z_1,z_2)$ are arbitrary solutions of the (1 + 1)-dimensional linear heat equation, $\theta ^k_1=\theta ^k_{22}$ , $k=0,\dots,n-1$ . Note that for $n=2$ , the first two families in this list have exactly the same representations as the first two families in the previous list, and

\begin{equation*}\mathfrak P^3\tilde {\theta }^1+\tilde {\theta }^0=4y\theta ^1_{22}+x^2\theta ^1_2-x\theta ^1+\theta ^0\end{equation*}

if $\tilde{\theta }^1=-\frac 43\theta ^1_1$ and $\tilde{\theta }^0=\theta ^0+4t^2\theta ^1_{12}+4tx\theta ^1_{22}+(x^2-2t)\theta ^1_2-2t\theta ^1_2-x\theta ^1$ .

Up to the $G^{\textrm{ess}}$ -equivalence, the generalised vector fields $ ((\mathfrak P^2+\varepsilon \mathfrak P^0)^nu )\partial _u$ , $ ((\mathfrak P^1)^nu )\partial _u$ and $ ((\mathfrak P^0)^nu )\partial _u$ exhaust those generalised symmetries of the equation (2) that are obtained using powers of operators associated with vector fields from $\mathfrak r$ . It is much more challenging to use vector fields from $\mathfrak{g}\setminus \mathfrak r$ for generalised reductions in a similar way. The question of whether operators associated with vector fields from $\mathfrak{g}\setminus \mathfrak r$ can generate solutions that are $G^{\textrm{ess}}$ -inequivalent to the above solutions needs a careful analysis.

9. On Kramers equations

An important subclass of the class $\bar{\mathcal{F}}$ of (1 + 2)-dimensional ultraparabolic Fokker–Planck equations, which are of the general form (1), is the class $\mathcal{K}$ of Kramers equations (more commonly called the Klein–Kramers equations)

\begin{equation*} u_t+xu_y=F(y)u_x+\gamma (xu+u_x)_x. \end{equation*}

These equations describe the evolution of the probability density function $u(t,x,y)$ of a Brownian particle in the phase space $(y,x)$ in one spatial dimension. Here, the variables $t$ , $y$ and $x$ play the role of the time, the position and the momentum, respectively, $F$ is an arbitrary smooth function of $y$ that is the derivative of the external potential with respect to $y$ , and $\gamma$ is an arbitrary nonzero constant, which is related to the friction coefficient. Up to a simple scale equivalence of Kramers equations, without loss of generality, one can assume $\gamma =1$ . The subclass $\mathcal{K}$ is singled out from the class $\bar{\mathcal{F}}$ by the auxiliary differential constraints $C=0$ , $A^0=A^1_x=A^2$ , $A^0_t=A^0_x=A^0_y=0$ and $A^1_t=0$ .

A preliminary group classification of the class $\mathcal{K}$ was presented in [Reference Spichak and Stogny37]. In particular, it was proved that the essential Lie invariance algebra of an equation from the class $\mathcal{K}$ is eight-dimensional if and only if $F(y)=ky$ up to shifts of $y$ , where $k=-\frac 34\gamma ^2$ or $k=\frac 3{16}\gamma ^2$ .

Any equation from the class $\bar{\mathcal{F}}$ with eight-dimensional essential Lie symmetry algebra is $\mathcal{G}_{\bar{\mathcal{F}}}$ -equivalent to the remarkable Fokker–Planck equation (2). (We will present the proof of this fact as well as the complete group classification of the class $\bar{\mathcal{F}}$ in a future paper.) Therefore, there exist point transformations that map the equations

(10) \begin{align} u_t+xu_y=\gamma u_{xx}+\gamma \left(x-\frac 34\gamma y\right)u_x+\gamma u,\end{align}

(11) \begin{align} u_t+xu_y=\gamma u_{xx}+\gamma \left(x+\frac 3{16}\gamma y\right)u_x+\gamma u \end{align}

to the equation (2).

Bases of the essential Lie invariance algebras $\mathfrak{g}^{\textrm{ess}}_{(10)}$ and $\mathfrak{g}^{\textrm{ess}}_{(11)}$ of the equations (10) and (11) consist of the vector fields

\begin{align*} &\hat{\mathcal{P}}^t=\textrm{e}^{-\gamma t}\left (\frac 1\gamma \partial _t-\frac 32y\partial _y+\frac 12(3\gamma y-x)\partial _x -\left(\frac 14x^2+\frac 34\gamma xy+\frac 9{16}\gamma ^2y^2-\frac 32\right)u\partial _u\right ),\\ &\hat{\mathcal{D}}=\frac 2\gamma \partial _t+u\partial _u,\quad \hat{\mathcal{K}}=\textrm{e}^{\gamma t}\left (\frac 1\gamma \partial _t+\frac 32y\partial _y+\frac 12(3\gamma y\!+\!x)\partial _x -\left(\frac 34 x^2\!+\!\frac 34\gamma xy\!-\!\frac 9{16}\gamma ^2y^2\!+\!\frac 12\right)u\partial _u\right ),\\ &\hat{\mathcal{P}}^3=\textrm{e}^{\frac 32\gamma t}\left (\frac 1\gamma \partial _y+\frac 32\partial _x+\frac 34(\gamma y-2x)u\partial _u\right ),\quad \hat{\mathcal{P}}^2=\textrm{e}^{\frac 12\gamma t}\left (\frac 1\gamma \partial _y+\frac 12\partial _x\right ),\\ &\hat{\mathcal{P}}^1=\textrm{e}^{-\frac 12\gamma t}\left (\frac 1\gamma \partial _y-\frac 12\partial _x+\frac 14(3\gamma y+2x)u\partial _u\right ),\quad \hat{\mathcal{P}}^0=\textrm{e}^{-\frac 32\gamma t}\left (\frac 1{\gamma }\partial _y-\frac 32\partial _x\right ),\quad \hat{\mathcal{I}}=u\partial _u\\[3pt] & \!\!\!\!\!\!\!\!\mbox{and}\qquad\\[3pt] &\check{\mathcal{P}}^t=\textrm{e}^{-\frac 12\gamma t}\left (\frac 1\gamma \partial _t-\frac 34y\partial _y+\frac 18(3\gamma y-2x)\partial _x+u\partial _u\right ),\quad \check{\mathcal{D}}=\frac 4\gamma \partial _t+2u\partial _u,\\ &\check{\mathcal{K}}=\textrm{e}^{\frac 12\gamma t}\left (\frac 4\gamma \partial _t+3 y\partial _y+\frac 12(3\gamma y+2x)\partial _x -\left(x^2+\frac 32\gamma xy+\frac 9{16}\gamma ^2y^2\right)u\partial _u\right ),\\ &\check{\mathcal{P}}^3=\textrm{e}^{\frac 34\gamma t}\left (\frac 8\gamma \partial _y+6\partial _x-\frac 32(4x+\gamma y)u\partial _u\right ),\quad \check{\mathcal{P}}^2=\textrm{e}^{\frac 14\gamma t}\left (\frac 4\gamma \partial _y+\partial _x-\left(x+\frac{3}{4}\gamma y\right)u\partial _u\right ),\\ &\check{\mathcal{P}}^1=\textrm{e}^{-\frac 14\gamma t}\left (\frac 2\gamma \partial _y-\frac{1}{2}\partial _x\right ),\quad \check{\mathcal{P}}^0=\textrm{e}^{-\frac 34\gamma t}\left (\frac 1\gamma \partial _y-\frac{3}{4}\partial _x\right ),\quad \check{\mathcal{I}}=u\partial _u, \end{align*}

respectively. We choose the basis elements of the algebra $\mathfrak{g}^{\textrm{ess}}_{(10)}$ (resp. $\mathfrak{g}^{\textrm{ess}}_{(11)}$ ) in such a way that the commutation relations between them coincide with those between the basis elements of the algebra $\mathfrak{g}^{\textrm{ess}}$ presented in Section 2. Thus, the fact that the algebras $\mathfrak{g}^{\textrm{ess}}$ , $\mathfrak{g}^{\textrm{ess}}_{(10)}$ and $\mathfrak{g}^{\textrm{ess}}_{(11)}$ are isomorphic becomes obvious.

Point transformations $\Phi _{(10)}$ and $\Phi _{(11)}$ that, respectively, map the equations (10) and (11) to the remarkable Fokker–Planck equation (2) are constructed using the conditions that $\Phi _{(10)*}\hat{\mathcal{V}}=\mathcal{V}$ and $\Phi _{(11)*}\check{\mathcal{V}}=\mathcal{V}$ for $\mathcal{V}\in \{\mathcal{P}^t, \mathcal{D}, \mathcal{K}, \mathcal{P}^3, \mathcal{P}^2, \mathcal{P}^1, \mathcal{P}^0, \mathcal{I}\}$ . Thus, the pushforwards $\Phi _{(10)*}$ and $\Phi _{(11)*}$ establish the isomorphisms of the algebras $\mathfrak{g}^{\textrm{ess}}_{(10)}$ and $\mathfrak{g}^{\textrm{ess}}_{(11)}$ to the algebra $\mathfrak{g}^{\textrm{ess}}$ , $\Phi _{(10)*}\mathfrak{g}^{\textrm{ess}}_{(10)}=\mathfrak{g}^{\textrm{ess}}$ and $\Phi _{(11)*}\mathfrak{g}^{\textrm{ess}}_{(11)\vphantom{\frac{1}{2}}}=\mathfrak{g}^{\textrm{ess}}$ . As a result, we obtain the point transformations

\begin{align*} &\Phi _{(10)}\,:\quad \tilde t=\textrm{e}^{\gamma t},\quad \tilde x=\textrm{e}^{\frac 12\gamma t}\left(\frac 32\gamma y+x\right),\quad \tilde y=\gamma \textrm{e}^{\frac 32\gamma t}y,\quad \tilde u=\textrm{e}^{-\frac{1}{4}(\frac{3}{2}\gamma y+x)^2-\frac{3}{2}\gamma t}u, \\[1ex] &\Phi _{(11)}\,:\quad \tilde t=2\textrm{e}^{\frac 12\gamma t},\quad \tilde x=\textrm{e}^{\frac 14\gamma t}\left(\frac 34\gamma y+x\right),\quad \tilde y=\gamma \textrm{e}^{\frac 34\gamma t}y,\quad \tilde u=\textrm{e}^{-\gamma t}u, \end{align*}

where the variables with tildes correspond to the equation (2), and the variables without tildes correspond to the source equation (10) or (11). By direct computation, we check that indeed the transformations $\Phi _{(10)}$ and $\Phi _{(11)}$ respectively map the equations (10) and (11) to the equation (2). In other words, an arbitrary solution $\tilde u=\,f(t,x,y)$ of (2) is pulled back by $\Phi _{(10)}$ and $\Phi _{(11)}$ to the solutions $u=\Phi _{(10)}^*\,f$ and $u=\Phi _{(11)}^*\,f$ of the equations (10) and (11), respectively,

\begin{align*} &u=\Phi _{(10)}^*\,f =\textrm{e}^{\frac 1{16}(3\gamma y+2x)^2+\frac 32\gamma t}f \left ( \textrm{e}^{\gamma t}, \textrm{e}^{\frac 12\gamma t}\left (\frac 32\gamma y+x\right ), \gamma \textrm{e}^{\frac 32\gamma t}y \right ), \\[1ex] &u=\Phi _{(11)}^*\,f =\textrm{e}^{\gamma t}f \left ( 2\textrm{e}^{\frac 12\gamma t}, \textrm{e}^{\frac 14\gamma t}\left (\frac 34\gamma y+x\right ), \gamma \textrm{e}^{\frac 34\gamma t}y \right ). \end{align*}

Using the families of solutions of the equation (2) that have been constructed in Sections 5 and 6, we find the following families of solutions for the equation (10):

\begin{align*} &u=\textrm{e}^{\frac 1{16}(3\gamma y+2x)^2+\frac{11}8\gamma t}\left|\frac 32\gamma y+x\right|^{-\frac 14} \theta ^\mu \left ( \frac 94\gamma \tilde{\varepsilon}\textrm{e}^{\frac 32\gamma t}y,\, \textrm{e}^{\frac 34\gamma t}\left|\frac 32\gamma y+x\right|^{\frac 32} \right )\ \text{with}\ \mu =\frac 5{36}, \ \tilde{\varepsilon}:=\mathop{\textrm{sgn}}\left(\!x+\frac32\gamma y\!\right)\!, \\[1ex] &u=\textrm{e}^{\gamma t}\,\theta \left ( \frac 13\textrm{e}^{3\gamma t}+2\varepsilon \textrm{e}^{\gamma t}-\textrm{e}^{-\gamma t},\, \textrm{e}^{-\frac 12\gamma t}\left (\frac 12\gamma (\textrm{e}^{2\gamma t}-3\varepsilon )y-(\textrm{e}^{2\gamma t}+\varepsilon )x\right ) \right ), \\[1ex] &u=\textrm{e}^{\frac 1{16}(3\gamma y+2x)^2+\frac 32\gamma t}\,\theta \left ( \frac 43\textrm{e}^{3\gamma t},\, \textrm{e}^{\frac 32\gamma t}(\gamma y+2x) \right ), \\[1ex] &u=\textrm{e}^{\frac 1{16}(3\gamma y+2x)^2+\frac 32\gamma t}\,\theta \left ( 4\textrm{e}^{\gamma t},\, \textrm{e}^{\frac 12\gamma t}(3\gamma y+2x) \right ), \\[1ex] &u=\textrm{e}^{\frac 12\tilde \zeta -\frac{x\zeta ^2}{24\gamma y}+\frac 32\gamma \kappa t}\zeta |y|^{\kappa -\frac 43} \left (C_1M\left (\kappa,\frac 43,\tilde \zeta \right )+C_2U\left (\kappa,\frac 43,\tilde \zeta \right )\right )\\ &\qquad \mbox{with}\quad \zeta \,:\!=3\gamma y+2x,\quad \tilde \zeta \,:\!=\frac 1{72}(\gamma y)^{-1}\zeta ^3. \end{align*}

In the same way, we can also obtain solutions of the equation (11),

\begin{align*} &u=\textrm{e}^{\frac{15}{16}\gamma t}\left|\frac 34\gamma y+x\right|^{-\frac 14}\,\theta ^\mu \left ( \frac{9}{4}\gamma \tilde{\varepsilon}\textrm{e}^{\frac 34\gamma t}y,\, \textrm{e}^{\frac 38\gamma t}\left|\frac 34\gamma y+x\right|^{\frac 32} \right )\quad \text{with}\quad \mu =\frac 5{36}, \ \tilde{\varepsilon}:=\mathop{\textrm{sgn}}\left(x+\frac34\gamma y\right),\\ &u=\textrm{e}^{-\frac 1{8}(\frac 34\gamma y+x)^2+\frac 34\gamma t}\,\theta \left ( \frac 83\textrm{e}^{\frac 32\gamma t}+4\varepsilon \textrm{e}^{\frac 12\gamma t}-\frac{1}{2}\textrm{e}^{-\frac 12\gamma t},\, \textrm{e}^{-\frac 14\gamma t}\left (\frac 18\gamma (4\textrm{e}^{\gamma t}-3\varepsilon )y-\frac{1}{2}(4\textrm{e}^{\gamma t}+\varepsilon )x\right ) \right ),\\ & u=\textrm{e}^{\gamma t}\,\theta \left( \frac{32}3\textrm{e}^{\frac 32\gamma t},\, \textrm{e}^{\frac 34\gamma t}(\gamma y+4x) \right),\qquad u=\textrm{e}^{\gamma t}\,\theta \left ( 32\textrm{e}^{\frac 12\gamma t},\, \textrm{e}^{\frac 14\gamma t}(3\gamma y+4x) \right ),\\ & u=\textrm{e}^{-\zeta ^3+\frac 14\gamma t(3\kappa +1)} \zeta |y|^{\kappa -1} \left (C_1M\left (\kappa,\frac 43,\zeta ^3\right )+C_2U\left (\kappa,\frac 43,\zeta ^3\right )\right )\\ &\qquad \mbox{with}\quad \zeta \,:\!=(9\gamma y)^{-\frac 13}\left (\frac 34\gamma y+x\right ). \end{align*}

Recall that $\theta ^\mu =\theta ^\mu (z_1,z_2)$ and $\theta =\theta (z_1,z_2)$ denote arbitrary solutions of the (1 + 1)-dimensional linear heat equation with potential $\mu z_2^{-2}$ and of the (1 + 1)-dimensional linear heat equation, $\theta ^\mu _1=\theta ^\mu _{22}+\mu z_2^{-2}\theta ^\mu$ and $\theta _1=\theta _{22}$ , respectively. $M(a,b,\omega )$ and $U(a,b,\omega )$ are the standard solutions of Kummer’s equation $\omega \varphi _{\omega \omega }+(b-\omega )\varphi _\omega -a\varphi =0$ .

Similarly to Section 8, we can also construct more general families of solutions of the equations (10) and (11), carrying out their generalised reductions or acting by Lie-symmetry operators on known solutions. At the same time, we can again use the transformations $\Phi _{(10)}$ and $\Phi _{(11)}$ just to pull back the solutions of the equation (2) that are obtained in Section 8.

Thus, the families of solutions of the equation (10) that are invariant with respect to the generalised symmetries $ ((\hat{\mathfrak P}^2+\varepsilon \hat{\mathfrak P}^0)^nu )\partial _u$ , $ ((\hat{\mathfrak P}^1)^nu )\partial _u$ or $ ((\hat{\mathfrak P}^0)^nu )\partial _u$ are, respectively, constituted by the solutions of the form

\begin{align*} &u=\sum _{k=0}^{n-1}(\hat{\mathfrak P}^1)^k\left (\textrm{e}^{\gamma t}\,\theta ^k(z_1,z_2)\right ) \\ &\qquad \mbox{with}\quad z_1=\frac 13\textrm{e}^{3\gamma t}+2\varepsilon \textrm{e}^{\gamma t}-\textrm{e}^{-\gamma t},\ \ z_2=\textrm{e}^{-\frac 12\gamma t}\left (\frac 12\gamma (\textrm{e}^{2\gamma t}-3\varepsilon )y-(\textrm{e}^{2\gamma t}+\varepsilon )x\right ), \\[1ex] &u=\sum _{k=0}^{n-1}(\hat{\mathfrak P}^2)^k \left ( \textrm{e}^{\frac 1{16}(3\gamma y+2x)^2+\frac 32\gamma t}\, \theta ^k(z_1,z_2) \right ) \quad \mbox{with}\quad z_1=\frac 43\textrm{e}^{3\gamma t},\ \ z_2=\textrm{e}^{\frac 32\gamma t}(\gamma y+2x), \\[1ex] &u=\sum _{k=0}^{n-1}(\hat{\mathfrak P}^3)^k \left ( \textrm{e}^{\frac 1{16}(3\gamma y+2x)^2+\frac 32\gamma t}\, \theta ^k(z_1,z_2) \right ) \quad \mbox{with}\quad z_1=4\textrm{e}^{\gamma t},\ \ z_2=\textrm{e}^{\frac 12\gamma t}(3\gamma y+2x). \end{align*}

Here and in what follows, $\theta ^k=\theta ^k(z_1,z_2)$ are arbitrary solutions of the (1 + 1)-dimensional linear heat equation, $\theta ^k_1=\theta ^k_{22}$ , $k=0,\dots,n-1$ . The differential operators in total derivatives

\begin{align*} &\hat{\mathfrak P}^3\,:\!=\textrm{e}^{\frac 32\gamma t}\left (\frac 1\gamma \textrm{D}_y+\frac 32\textrm{D}_x-\frac 34(\gamma y-2x)\right ),\\[1ex] &\hat{\mathfrak P}^2\,:\!=\textrm{e}^{\frac 12\gamma t}\left (\frac 1\gamma \textrm{D}_y+\frac 12\textrm{D}_x\right ),\\[1ex] &\hat{\mathfrak P}^1\,:\!=\textrm{e}^{-\frac 12\gamma t}\left (\frac 1\gamma \textrm{D}_y-\frac 12\textrm{D}_x-\frac 14(3\gamma y+2x)\right ),\\[1ex] &\hat{\mathfrak P}^0\,:\!=\textrm{e}^{-\frac 32\gamma t}\left (\frac 1{\gamma }\textrm{D}_y-\frac 32\textrm{D}_x\right ) \end{align*}

are Lie-symmetry operators of the equation (10) and are associated with the Lie-symmetry vector fields $-\hat{\mathcal{P}}^3$ , $-\hat{\mathcal{P}}^2$ , $-\hat{\mathcal{P}}^1$ , $-\hat{\mathcal{P}}^0$ , respectively.

Analogously, the solutions of the equation (11) that are invariant with respect to the generalised symmetries $ ((\check{\mathfrak P}^2+\varepsilon \check{\mathfrak P}^0)^nu )\partial _u$ , $ ((\check{\mathfrak P}^1)^nu )\partial _u$ or $ ((\check{\mathfrak P}^0)^nu )\partial _u$ are of the form

\begin{align*} &u=\sum _{k=0}^{n-1}(\check{\mathfrak P}^1)^k\left ( \textrm{e}^{-\frac 1{8}(\frac 34\gamma y+x)^2+\frac 34\gamma t}\,\theta ^k(z_1,z_2)\right ) \\ &\qquad \mbox{with}\quad z_1=\frac 83\textrm{e}^{\frac 32\gamma t}+4\varepsilon \textrm{e}^{\frac 12\gamma t}-\frac{1}{2}\textrm{e}^{-\frac 12\gamma t},\ \ z_2=\textrm{e}^{-\frac 14\gamma t}\left (\frac 18\gamma (4\textrm{e}^{\gamma t}-3\varepsilon )y-\frac12(4\textrm{e}^{\gamma t}+\varepsilon )x\right ), \\[1ex] &u=\sum _{k=0}^{n-1}(\check{\mathfrak P}^2)^k \left (\textrm{e}^{\gamma t}\,\theta ^k(z_1,z_2)\right ) \quad \mbox{with}\quad z_1=\frac{32}3\textrm{e}^{\frac 32\gamma t},\ \ z_2= \textrm{e}^{\frac 34\gamma t}(\gamma y+4x), \\[1ex] &u=\sum _{k=0}^{n-1}(\check{\mathfrak P}^3)^k \left ( \textrm{e}^{\gamma t}\,\theta ^k(z_1,z_2) \right ) \quad \mbox{with}\quad z_1=32\textrm{e}^{\frac 12\gamma t},\ \ z_2=\textrm{e}^{\frac 14\gamma t}(3\gamma y+4x), \end{align*}

where the differential operators in total derivatives

\begin{align*} &\check{\mathfrak P}^3\,:\!=\textrm{e}^{\frac 34\gamma t}\left (\frac 8\gamma \textrm{D}_y+6\textrm{D}_x+\frac 32(4x+\gamma y)\right ),\quad \check{\mathfrak P}^2\,:\!=\textrm{e}^{\frac 14\gamma t}\left (\frac 4\gamma \textrm{D}_y+\textrm{D}_x+x+\frac{3}{4}\gamma y\right ),\\[1ex] &\check{\mathfrak P}^1\,:\!=\textrm{e}^{-\frac 14\gamma t}\left (\frac 2\gamma \textrm{D}_y-\frac{1}{2}\textrm{D}_x\right ),\quad \check{\mathfrak P}^0\,:\!=\textrm{e}^{-\frac 34\gamma t}\left (\frac 1\gamma \textrm{D}_y-\frac{3}{4}\textrm{D}_x\right ) \end{align*}

are Lie-symmetry operators of the equation (11) and are associated with the Lie-symmetry vector fields $-\check{\mathcal{P}}^3$ , $-\check{\mathcal{P}}^2$ , $-\check{\mathcal{P}}^1$ , $-\check{\mathcal{P}}^0$ .

Note that $\Phi _{(10)*}\hat{\mathfrak Q}=\mathfrak Q$ and $\Phi _{(11)*}\check{\mathfrak Q}=\mathfrak Q$ for $\mathfrak Q\in \{\mathfrak P^3, \mathfrak P^2, \mathfrak P^1,\mathfrak P^0,1\}$ .