Note on a ball rolling over a sphere: integrable Chaplygin system with an invariant measure without Chaplygin Hamiltonization

In this note we consider the nonholonomic problem of rolling without slipping and twisting of an $n$-dimensional balanced ball over a fixed sphere. This is a $SO(n)$–Chaplygin system with an invariant measure that reduces to the cotangent bundle $T^*S^{n-1}$. For the rigid body inertia operator $\mathbb I\omega=I\omega+\omega I$, $I=\operatorname{diag}(I_1,\dots,I_n)$ with a symmetry $I_1=I_2=\dots=I_{r} \ne I_{r+1}=I_{r+2}=\dots=I_n$, we prove that the reduced system is integrable, general trajectories are quasi-periodic, while for $r\ne 1,n-1$ the Chaplygin reducing multiplier method does not apply.