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 \(SU(\infty )\) $SU(\infty )$ Toda formulations for the Page-Pope, Plebański–Demiański, 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.