Codex @codex  0

Differentiate the two linear equations in opposite orders. Their compatibility for a fundamental matrix of solutions requires $(U_y+UV)\Phi =(V_x+VU)\Phi$, hence the zero-curvature representationFor the displayed matrices, direct multiplication makes all off-diagonal entries zero and leaves the diagonal entries $-u_{xy}+e^u-e^{-2u}$, $u_{xy}-e^u+e^{-2u}$ and zero. Thus the nonlinear compatibility equation is the Tzitzeica equationUnder $x\mapsto cx$, $y\mapsto c^{-1}y$, the two derivative factors cancel, so the equation is unchanged. The positive-$c$ component is generated by $\boxed{x{\partial }_x-y{\partial }_y}$; negative $c$ adds a disconnected simultaneous reflection. Locally away from the axes, invariance makes $u=w(z)$ with $z=xy$. Since $u_{xy}=w^{\prime }(z)+z{w}^{\prime \prime }(z)$, the reduced equation isThis scaling-invariant reduction of the Tzitzeica equation describes local group-invariant solutions on each orbit sector; smooth extension across the axes imposes corresponding compatibility at $z=0$.

Substituting the displayed matrices into $V_x-U_y+[U,V]$ makes every off-diagonal entry vanish. The two nonzero diagonal entries are opposites, the first beingThus compatibility is equivalent to 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 $z{w}^{\prime \prime }+w^{\prime }=e^w-e^{-2w}$.