Solution
= Solution
For Maxwell theory the group is Abelian, so write $J=e^\chi$. The reduced equation loses its commutators and becomes
$$
\partial_w\partial_{\bar w}\chi
+\partial_z\partial_{\bar z}\chi=0.
$$
Since the Euclidean <Laplacian> is
$$
\Delta_{\mathbb R^4}=4(\partial_w\partial_{\bar w}
+\partial_z\partial_{\bar z}),
$$
the anti-self-dual Maxwell equations in this gauge are equivalent to
$$
\boxed{\Delta_{\mathbb R^4}\chi=0}.
$$