  We study the Hamilton-Jacobi and massive Klein-Gordon equations in the
general Kerr-(Anti) de Sitter black hole background in all dimensions. Complete
separation of both equations is carried out in cases when there are two sets of
equal black hole rotation parameters. We analyze explicitly the symmetry
properties of these backgrounds that allow for this Liouville integrability and
construct a nontrivial irreducible Killing tensor associated with the enlarged
symmetry group which permits separation. We also derive first-order equations
of motion for particles in these backgrounds and examine some of their
properties. This work greatly generalizes previously known results for both the
Myers-Perry metrics, and the Kerr-(Anti) de Sitter metrics in higher
dimensions.
