  The Kerr spacetime in Kaluza-Klein theory describes a rotating black hole in
four dimensions from the Kaluza-Klein point of view and involves the signature
of an extra dimension that shows up through the appearance of the electric and
dilaton charges. In this paper, we study the separability properties of the
Hamilton-Jacobi equation for geodesics and the associated hidden symmetries in
the spacetime of the Kerr-Kaluza-Klein black hole. We show that the complete
separation of variables occurs only for massless geodesics, implying the
existence of hidden symmetries generated by a second rank conformal Killing
tensor. Employing a simple procedure built up on an "effective" metric, which
is conformally related to the original spacetime metric and admits a complete
separability structure, we construct the explicit expression for the conformal
Killing tensor. Next, we study the properties of the geodesic motion in the
equatorial plane, focusing on the cases of static and rotating Kaluza-Klein
black holes separately. In both cases, we obtain the defining equations for the
boundaries of the regions of existence, boundedness and stability of the
circular orbits as well as the analytical formulas for the orbital frequency,
the radial and vertical epicyclic frequencies of the geodesic motion.
Performing a detailed numerical analysis of these equations and frequencies, we
show that the physical effect of the extra dimension amounts to the significant
enlarging of the regions of existence, boundedness and stability towards the
event horizon, regardless of the classes of orbits.
