  The remarkable and unexpected separability of the Hamilton-Jacobi and
Klein-Gordon equations in the background of a rotating four-dimensional black
hole played an important role in the construction of generalisations of the
Kerr metric, and in the uncovering of hidden symmetries associated with the
existence of Killing tensors. In this paper, we show that the Hamilton-Jacobi
and Klein-Gordon equations are separable in Kerr-AdS backgrounds in all
dimensions, if one specialises the rotation parameters so that the metrics have
cohomogeneity 2. Furthermore, we show that this property of separability
extends to the NUT generalisations of these cohomogeneity-2 black holes that we
obtained in a recent paper. In all these cases, we also construct the
associated irreducible rank-2 Killing tensor whose existence reflects the
hidden symmetry that leads to the separability. We also consider some
cohomogeneity-1 specialisations of the new Kerr-NUT-AdS metrics, showing how
they relate to previous results in the literature.
