From 2D Integrable Systems to Self-Dual Gravity

We explain how to construct solutions to the self-dual Einstein vacuum equations from solutions of various two-dimensional integrable systems by exploiting the fact that the Lax formulations of both systems can be embedded in that of the self-dual Yang--Mills equations. We illustrate this by constructing explicit self-dual vacuum metrics on $\R^2\times \Sigma$, where $\Sigma$ is a homogeneous space for a real subgroup of $SL(2, \C)$ associated with the two-dimensional system.