Hidden symmetries of generalised gravitational instantons

For conformally Kähler Riemannian four-manifolds with a Killing field, we present a framework to solve the field equations for generalised gravitational instantons corresponding to conformal self-duality and to cosmological Einstein-Maxwell. After deriving generic identities for the curvature of such manifolds without assuming field equations, we obtain $SU(\mathrm{\infty })$ Toda formulations for the Page-Pope, Plebanski-Demianski, and Chen-Teo classes, we show how to solve the (modified) Toda equation, and we use this to find conformally self-dual and Einstein-Maxwell generalisations of these geometries.