What is a completely integrable nonholonomic dynamical system?

We compare the geometry of a toral fibration defined by the common level sets of the integrals of a Liouville integrable Hamiltonian system with a toral fibration coming from a completely integrable nonholonomic system. We illustrate their differences using the following examples: the nonholonomic oscillator, Chaplyginʹs skate, Routhʹs sphere and the rolling oblate ellipsoid of revolution.