Integrability versus separability for the multi-centre metrics

TL;DR

This paper investigates the conditions under which multi-centre metrics, solutions to Einstein's equations with self-dual curvature, exhibit integrability and separability, revealing new classes of integrable systems and associated geometric structures.

Contribution

It identifies criteria for extra conserved quantities in multi-centre metrics and introduces new integrable subclasses with separable Hamilton-Jacobi equations and Killing-Yano tensors.

Findings

Certain multi-centre metrics admit quadratic conserved quantities.

New integrable subclasses are characterized by specific geometric properties.

Construction of new Killing-Yano tensors within these subclasses.

Abstract

The multi-centre metrics are a family of euclidean solutions of the empty space Einstein equations with self-dual curvature. For this full class, we determine which metrics do exhibit an extra conserved quantity quadratic in the momenta, induced by a Killing-St\" ackel tensor. Our systematic approach brings to light a subclass of metrics which correspond to new classically integrable dynamical systems. Within this subclass we analyze on the one hand the separation of coordinates in the Hamilton-Jacobi equation and on the other hand the construction of some new Killing-Yano tensors.