Gravitational instantons, confocal quadrics and separability of the Schrödinger and Hamilton–Jacobi equations

Abstract

A hyper-Kähler 4-metric with a triholomorphic SU(2) action gives rise to a family of confocal quadrics in Euclidean 3-space when cast in the canonical form of a hyper-Kähler 4-metric with a triholomorphic circle action. Moreover, at least in the case of geodesics orthogonal to the U(1) fibres, both the covariant Schrödinger and the Hamilton–Jacobi equations are separable and the system integrable.