# Conformal geodesics on gravitational instantons

## TL;DR

This paper investigates the integrability of conformal geodesic flows on gravitational instantons, providing new examples of integrable systems and analyzing specific cases like Taub NUT, Eguchi-Hanson, and Fubini-Study metrics.

## Contribution

It demonstrates the integrability of conformal geodesic flow on certain gravitational instantons, including the first example on a non-symmetric space, and analyzes specific cases with explicit solutions.

## Key findings

- Complete integration of conformal geodesics on anti-self-dual Taub NUT instanton.
- Characterization of conformal geodesics on Eguchi-Hanson orbits.
- Use of conformal Killing-Yano tensors to establish integrability on Fubini-Study metric.

## Abstract

We study the integrability of the conformal geodesic flow (also known as the conformal circle flow) on the $SO(3)$--invariant gravitational instantons. On a hyper--K\"ahler four--manifold the conformal geodesic equations reduce to geodesic equations of a charged particle moving in a constant self--dual magnetic field. In the case of the anti--self--dual Taub NUT instanton we integrate these equations completely by separating the Hamilton--Jacobi equations, and finding a commuting set of first integrals. This gives the first example of an integrable conformal geodesic flow on a four--manifold which is not a symmetric space. In the case of the Eguchi--Hanson we find all conformal geodesics which lie on the three--dimensional orbits of the isometry group. In the non--hyper--K\"ahler case of the Fubini--Study metric on $\CP^2$ we use the first integrals arising from the conformal Killing--Yano tensors to recover the known complete integrability of conformal geodesics.