  Bifibrations, in symplectic geometry called also dual pairs, play a relevant
role in the theory of superintegrable Hamiltonian systems. We prove the
existence of an analogous bifibrated geometry in dynamical systems with a
symmetry group such that the reduced dynamics is periodic. The integrability of
such systems has been proven by M. Field and J. Hermans with a reconstruction
technique. We apply the result to the nonholonomic system of a ball rolling on
a surface of revolution.
