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

Geometry of Invariant Tori of Certain Integrable Systems with Symmetry and an Application to a Nonholonomic System