Conformal geodesics on gravitational instantons

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-Kahler 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-Kahler case of the Fubini-Study metric on CP2 we use the first integrals arising from the conformal Killing-Yano tensors to recover the known complete integrability of conformal geodesics.