= Euclidean complex-coordinate anti-self-dual Lax pair
{c}
{title2=$L=D_w-\lambda D_{\bar z},\quad M=D_z+\lambda D_{\bar w}$}
For metric $2(dw\,d\bar w+dz\,d\bar z)$, the <commutator> is $[L,M]=F_{wz}+\lambda(F_{w\bar w}+F_{z\bar z})+\lambda^2F_{\bar w\bar z}$. Vanishing for every <spectral parameter> $\lambda$ is exactly the <Anti-self-dual Yang-Mills equations>. The derivative parts span a totally null two-plane in the complexified tangent space. Thus an invertible fundamental solution of $L\Psi=M\Psi=0$ exists locally exactly when the three <gauge curvature> coefficients vanish; compatibility of one possibly zero vector alone is not enough.
// Destination: fiber-bundle.bigb
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