Solution
= Solution
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 being
$$
-u_{xy}+e^u-e^{-2u}.
$$
Thus compatibility is equivalent to the <Tzitzeica equation>
$$
\boxed{u_{xy}=F(u)=e^u-e^{-2u}}.
$$