Nonholonomic connections, time reparametrizations, and integrability of the rolling ball over a sphere

We study a time reparametrisation of the Newton type equations on Riemannian manifolds slightly modifying the Chaplygin multiplier method, allowing us to consider the Chaplygin method and the Maupertuis principle within a unified framework. As an example, the reduced nonholonomic problem of rolling without slipping and twisting of an $n$-dimensional balanced ball over a fixed sphere is considered. For a special inertia operator (depending on $n$ parameters) we prove complete integrability when the radius of the ball is twice the radius of the sphere. In the case of $SO(l)\times SO(n-l)$ symmetry, noncommutative integrability for any ratio of the radii is established.