MOA2DIS mother of integrability

Why self dual yang mills in 4 dimensions is the Mother Of All 2d Integrable Systems. At least that is what Ketov says in his last chapter.

Found the sources
http://arxiv.org/pdf/hep-th/9307021v2.pdf
http://ac.els-cdn.com/037596018990964X/1-s2.0-037596018990964X-main.pdf?_tid=410488d6-3295-11e2-8bd0-00000aab0f6b&acdnat=1353362920_03278f52f496a8df2a191a8dae74b67f

Let's work on (2,2) signature  with metric  . Look at the self-dual Yang-Mills equations on it. In these coordinates they read:




Calling the components of the gauge field A,B,C,D respectively. We can ask that they don't depend on u or y. We can also ask that A=D, but I don't know why you would ask such a thing.




We have put three equations into 2 by inserting a  and asking to read of the coefficients of the commutator of the above operators.

The quadratic piece demands that B not depend on x. There are three possibilities for B after gauge transforming. Let us pick the one where B is proportional to the generator for the Cartan in it's usual form multiplied by some t dependent function.

The equations are solved if we write the following for A and C.




and



The second choice above gives the nonlinear Schrodinger equation with attractive self interaction.

We made several choices along the way, other choices would have resulted in NLS with repulsive interaction or KdV.

If that is all there is to it, the title MOA2DIS seems to be excessive.