Integrability, Monodromy Evolving Deformations, and Self-Dual Bianchi IX Systems

We derive a nonlinear system of ODE's related to complex Bianchi IX metrics with self-dual Weyl curvature from the compatibility conditions of a novel type of monodromy evolving linear system. The analysis of the linear system yields a nontrivial separation of variables leading to the general solution of the nonlinear equations. In general, the solution is densely branched, but we find a single valued family of special solutions corresponding to the self-dual Bianchi IX, vacuum Einstein equations. These nonlinear equations also arise in fluid dynamics and in two-dimensional topological field theories.