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 × Σ, where Σ is a homogeneous space for a real subgro...

Proof. We use induction on n. For n = 1, the result is immediate, First, observe there exist linear transformations Ui (i = 1, 2): X + X such that (U,x, y) = &(x, y). For define&: X -R by Liz(y) = #Q(x, y). Then L,, is a linear functional and since X is an inner product space, there exists Z< E X with L,,(y) = (zi, y) by the canonical isomorphism between X and its dual. Define the transformatio...