Geodesics on metrics of Eguchi-Hanson type

Geodesic equations are solved when at least two of $\theta$, $\phi$ and $\psi$ are constant, or $r$ is constant, on scalar flat metrics of Eguchi-Hanson type. They can also be solved also on Eguchi-Hanson metrics which are Ricci flat if only $\phi$ is constant. However, the explicit solution of the geodesic equations is not available yet if only $\psi$ is constant.