  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.
