Hidden symmetries of near-horizon extremal Kerr-AdS-NUT geometries

We study the hidden symmetries, the symmetries associated with Killing tensors, of the near-horizon geometry of odd-dimensional Kerr-AdS-NUT black holes in two limits: generic extremal and extremal vanishing horizon (EVH) limits. Starting from a Kerr-AdS-NUT black hole in ellipsoidal coordinates which admit integrable geodesic equations, we obtain the near-horizon extremal and EVH geometries and their principal and Killing tensors by taking the near-horizon limit. We explicitly demonstrate that geodesic equations are separable and integrable on these near-horizon geometries. We also compute the constants of motion and read the Killing tensors of these near-horizon geometries from the constants of motion. As expected, they are the same as the Killing tensors given by taking the near-horizon limit.