Scaling-invariant reduction of the Tzitzeica equation

The Tzitzeica equation $u_{xy}=e^u-e^{-2u}$ is unchanged by $(x,y)\mapsto (cx,c^{-1}y)$. The identity component has generator $x{\partial }_x-y{\partial }_y$. Locally its invariant solutions are $u=w(xy)$, reducing the equation to $z{w}^{\prime \prime }+w^{\prime }=e^w-e^{-2w}$.