Complex hyperbolic 2-orbifolds with isolated singularities

For each prime $p$, this paper constructs compact complex hyperbolic $2$-manifolds with an isometric action of $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ that is not free and has only isolated fixed points. The case $p=2$ is special, and finding general examples for $p=2$ is related to whether or not complex hyperbolic lattices are conjugacy separable on torsion.