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We study a time reparametrisation of the Newton type equations on Riemannianmanifolds slightly modifying the Chaplygin multiplier method, allowing us toconsider the Chaplygin method and the Maupertuis principle within a unifiedframework. As an example, the reduced nonholonomic problem of rolling withoutslipping and twisting of an $n$-dimensional balanced ball over a fixed sphereis considered. For a special inertia operator (depending on $n$ parameters) weprove complete integrability when the radius of the ball is twice the radius ofthe sphere. In the case of $SO(l)\times SO(n-l)$ symmetry, noncommutativeintegrability for any ratio of the radii is established.

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