Geodesics on metrics of self-dual Taub-Nut type

Geodesic equations are solved when at least two of $\tau$, $\theta$, $\phi$ are constant on metrics of self-dual Taub-NUT type. They can also be solved also on self-dual Taub-NUT metrics if only $r$, $\theta$ or $\phi$ is constant. However, the explicit solution of the geodesic equations is not available yet if only $\tau$ is constant.