Lax pair for the anti-self-dual Yang-Mills equations
= Lax pair for the anti-self-dual Yang-Mills equations
{c}
{title2=ASDYM Lax pair}
For covariant derivatives on Euclidean four-space, one anti-self-dual Yang-Mills Lax pair is
$$
L=D_1+iD_2-\zeta(D_3-iD_4),
\qquad
M=D_3+iD_4+\zeta(D_1-iD_2).
$$
The condition $[L,M]=0$ for every $\zeta\in\mathbb{CP}^1$ is equivalent to anti-self-duality of the curvature.