Euclidean complex-coordinate anti-self-dual Lax pair ($L=D_w-\lambda D_{\bar{z}},M=D_z+\lambda D_{\bar{w}}$)

For metric $2(dwd\bar{w}+dzd\bar{z})$, the commutator is $[L,M]=F_{wz}+\lambda (F_{w\bar{w}}+F_{z\bar{z}})+{\lambda }^2{F}_{\bar{w}\bar{z}}$. 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