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The aim of this article is to extend the scope of the theory of regularity structures in order to deal with a large class of singular stochastic partial differential equations of the form
\begin{equation*}\partial_t u = \mathfrak{L} u+ F(u, \xi),\end{equation*}
where the differential operator 
$\mathfrak{L}$ fails to be elliptic. This is achieved by interpreting the base space 
$\mathbb{R}^{d}$ as a non-trivial homogeneous Lie group 
$\mathbb{G}$ such that the differential operator 
$\partial_t -\mathfrak{L}$ becomes a translation invariant hypoelliptic operator on 
$\mathbb{G}$. Prime examples are the kinetic Fokker-Planck operator 
$\partial_t -\Delta_v - v\cdot \nabla_x$ and heat-type operators associated with sub-Laplacians. As an application of the developed framework, we solve a class of parabolic Anderson type equations
\begin{equation*}\partial_t u = \sum_{i} X^2_i u + u (\xi-c)\end{equation*}
on the compact quotient of an arbitrary Carnot group.
