= Solution
The system prescribes both first derivatives of the same vector $\psi$, so it is overdetermined: the two mixed derivatives must agree. Differentiating gives
$$
\begin{aligned}
\partial_{\bar z}\partial_z\psi
&=(\partial_{\bar z}U)\psi+U\partial_{\bar z}\psi
=(\partial_{\bar z}U+UV)\psi,\\
\partial_z\partial_{\bar z}\psi
&=(\partial_zV)\psi+V\partial_z\psi
=(\partial_zV+VU)\psi.
\end{aligned}
$$
Thus every solution satisfies
$$
\left(
\partial_{\bar z}U-\partial_zV+[U,V]
\right)\psi=0.
$$
For the system to possess arbitrary nontrivial local solution data, the coefficient matrix must vanish:
$$
\boxed{\partial_{\bar z}U-\partial_zV+[U,V]=0}.
$$
This is the <zero-curvature condition>, or the <compatibility condition for an overdetermined linear system>.
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