= Solution
Substitute
$$
U=\frac12
\begin{pmatrix}
u_z&\lambda^{-1}e^{-u}\\
\lambda^{-1}e^u&-u_z
\end{pmatrix},
\qquad
V=\frac12
\begin{pmatrix}
-u_{\bar z}&\lambda e^u\\
\lambda e^{-u}&u_{\bar z}
\end{pmatrix}.
$$
The off-diagonal entries of
$$
U_{\bar z}-V_z+[U,V]
$$
cancel identically. Its upper-left entry is
$$
u_{z\bar z}
+\frac14(e^{-2u}-e^{2u})
=u_{z\bar z}-\frac12\sinh(2u),
$$
and its lower-right entry is the negative of this. Hence compatibility is equivalent to
$$
\boxed{u_{z\bar z}=\frac12\sinh(2u)}.
$$
This is the <Sinh-Gordon equation> in complex light-cone coordinates.
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