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–Kähler 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–Kähler case of the Fubini–Study metric on $\mathbb{CP}^2$ we use the first integrals arising from the conformal Killing–Yano tensors to recover the known complete integrability of conformal geodesics.

我们研究了共形测地线流（也称为共形圆流）在SO (3) 不变的引力瞬子上的可积性。在超凯勒四流形上，共形测地线方程简化为带电粒子在恒定自对偶磁场中运动的测地线方程。在反自对偶 Taub NUT 瞬子的情况下，我们通过分离 Hamilton-Jacobi 方程并找到一组对易的第一积分来完全整合这些方程。这给出了在非对称空间的四流形上的可积分共形测地线流的第一个示例。在 Eguchi-Hanson 的情况下，我们发现所有保形测地线都位于等距群的三维轨道上。在 Fubini-Study 度量的非超 Kähler 案例中$\mathbb{CP}^2$我们使用由保形 Killing-Yano 张量产生的第一个积分来恢复保形测地线的已知完全可积性。