Aspects of integrability in self-dual Einstein metrics and related equations (Q1819761)

The system of nonlinear partial differential equations describing complex self-dual Einstein metrics is shown to be equivalent to a system of linear differential equations (for certain potentials) and a nonlinear constraint. This linear problem has a structure very similar to that of the self-dual Yang-Mills equations. More general systems of equations are considered, e.g. those obtained by dropping the above mentioned constraint. All these equations can be formulated in terms of a Riemann- Hilbert-problem or a twistor construction, and can be interpreted as a dynamical motion in an infinite dimensional Grassmann manifold. It turns out that the group-theoretical structures of the self-dual Einstein equation are different from those of the classical examples of nonlinear integrable systems.