Higher Conserved Quantities for the Multi-Centre Metrics

The integrability of the geodesic flow for the multi-centre metrics began with the discovery of the generalized Runge–Lenz vector for the Taub–NUT metric [1] and the derivation of its Killing–Stäckel tensor in [2]. It was generalized to the Eguchi–Hanson metric in [3] where the Hamilton–Jacobi equation was separated. A further progress led to the integrability proof of the full 2-centre metric [2] which includes Taub–NUT and Eguchi–Hanson as particular cases. Despite these successes, a systematic analysis of the full family of the multi-centre metrics was still lacking. We will present a new approach which will determine which metrics, in this class, do exhibit a quadratic Killing–Stäckel tensor leading to classical integrability.