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

We study hidden symmetries, the symmetries associated with the Killing tensors, of the near horizon geometry of odd-dimensional Kerr-AdS-NUT black hole in two limits: generic extremal and extremal vanishing horizon (EVH) limits. Starting from Kerr-AdS-NUT black hole in ellipsoidal coordinates which admits integrable geodesic equations, we obtain the near horizon extremal/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 theses 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 we expected, they are the same as the Killing tensors given by taking the near horizon limit.