  We study the integrability of geodesic flow in the background of some
recently discovered charged rotating solutions of supergravity in four and five
dimensions. Specifically, we work with the gauged multicharge
Taub-NUT-Kerr-(Anti) de Sitter metric in four dimensions, and the $U(1)^3$
gauged charged-Kerr-(Anti) de Sitter black hole solution of N = 2 supergravity
in five dimensions. We explicitly construct the Killing tensors that permit
separation of the Hamilton-Jacobi equation in these spacetimes. These results
prove integrability for a large class of previously known supergravity
solutions, including several BPS solitonic states. We also derive first-order
equations of motion for particles in these backgrounds and examine some of
their properties. Finally, we also examine the Klein-Gordon equation for a
scalar field in these spacetimes and demonstrate separability.
