  We show that the conformal structure for the Riemannian analogues of Kerr
black-hole metrics can be given an ambitoric structure. We then discuss the
properties of the moment maps. In particular, we observe that the moment map
image is not locally convex near the singularity corresponding to the ring
singularity in the interior of the black hole. We then proceed to classify
regular ambitoric $4$-orbifolds with some completeness assumptions. The tools
developed also allow us to prove a partial classification of compact Riemannian
$4$-manifolds which admit a Killing $2$-form.
