= Solution
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 representation>
$$
\boxed{U_y-V_x+[U,V]=0.}
$$
For 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 equation>
$$
\boxed{u_{xy}=e^u-e^{-2u}.}
$$
Under $x\mapsto cx$, $y\mapsto c^{-1}y$, 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 $u_{xy}=w'(z)+zw''(z)$, the reduced equation is
$$
\boxed{zw''+w'=e^w-e^{-2w}.}
$$
This <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 $z=0$.
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