Solution
= Solution
In this gauge, $F_{\bar w\bar z}=0$ says that the remaining partial connection is flat. Hence locally
$$
A_{\bar w}=J^{-1}\partial_{\bar w}J,
\qquad
A_{\bar z}=J^{-1}\partial_{\bar z}J,
$$
or
$$
\boxed{A=J^{-1}\partial_{\bar w}J\,d\bar w
+J^{-1}\partial_{\bar z}J\,d\bar z}.
$$
The remaining curvature equation then becomes
$$
\boxed{\partial_w(J^{-1}\partial_{\bar w}J)
+\partial_z(J^{-1}\partial_{\bar z}J)=0}.
$$