Aspects of Integrability In Self-Dual Einstein Metrics and Related Equations

Abstract

The nonlinear system describing self-dual Einstein metrics and its generalizations are discussed from the point of view of integrability. It is shown that these nonlinear systems share a variety of remarkable features (such as the existence of a linear scattering problem, a group-theoretical solution technique similar to the Riemann-Hilbert problem, and a geometric interpretation as dynamical motion in an infinite dimensional Grassmann manifold) with nonlinear integrable systems known until now. Differences of the relevant group-theoretical structures between these two classes of nonlinear systems are also pointed out. These results lead to the conclusion that the nonlinear systems in question do form a new class of nonlinear integrable systems.