Under certain conditions the motion of superconducting vortices is primarily governed by an instability. We consider an averaged model, for this phenomenon, describing the motion of large numbers of such vortices. The model equations are parabolic, and, in one spatial dimension $x$, take the form \begin{eqnarray*} {H_2}_t &=& \frac{\partial}{\partial x} ( | {H_3} {H_2}_x-{H_2} {H_3}_x | {H_2}_x ), \\ {H_3}_t &=& \frac{\partial}{\partial x} ( | {H_3} {H_2}_x-{H_2} {H_3}_x | {H_3}_x ). \end{eqnarray*} where $H_2$ and $H_3$ are components of the magnetic field in the $y$ and $z$ directions respectively. These equations have an extremely rich group of symmetries and a correspondingly large class of similarity reductions. In this work, we look for non-trivial steady solutions to the model, deduce their stability and use a numerical method to calculate time-dependent solutions. We then apply Lie Group based similarity methods to calculate a complete catalogue of the model's similarity reductions and use this to investigate a number of its physically important similarity solutions. These describe the short time response of the superconductor as a current or magnetic field is switched on (or off).