First Integrals from Conformal Symmetries: Darboux-Koenigs Metrics and Beyond

TL;DR

This paper explores how conformal symmetries in conformally flat spaces can be used to generate higher order integrals for geodesic equations, providing new insights into Darboux-Koenigs metrics and their quantum analogues.

Contribution

It introduces a novel method leveraging conformal symmetries to derive higher order integrals and offers a new derivation of Darboux-Koenigs metrics, including quantum extensions.

Findings

Derived higher order integrals from conformal symmetries.

Provided a new derivation of Darboux-Koenigs metrics.

Extended the approach to quantum analogues.

Abstract

On spaces of constant curvature, the geodesic equations automatically have higher order integrals, which are just built out of first order integrals, corresponding to the abundance of Killing vectors. This is no longer true for general conformally flat spaces, but in this case there is a large algebra of conformal symmetries. In this paper we use these conformal symmetries to build higher order integrals for the geodesic equations. We use this approach to give a new derivation of the Darboux-Koenigs metrics, which have only one Killing vector, but two quadratic integrals. We also consider the case of possessing one Killing vector and two cubic integrals. The approach allows the quantum analogue to be constructed in a simpler manner.