Separability and hidden symmetries of Kerr-Taub-NUT spacetime in Kaluza-Klein theory

The Kerr-Taub-NUT spacetime in the Kaluza-Klein theory represents a localized stationary and axisymmetric object in four dimensions from the Kaluza-Klein viewpoint. That is, it harbors companion electromagnetic and dilaton fields, thereby showing up the signature of the extra fifth dimension. We explore the separability structure of this spacetime and show that the Hamilton-Jacobi equation for geodesics admits complete separation of variables only for massless geodesics. This implies the existence of hidden symmetries in the spacetime, which are generated by a conformal Killing tensor. Using a simple trick built up on a conformally related metric (an "effective" metric) with the Killing tensor, we construct an explicit expression for the conformal Killing tensor.