Аннотация:
We consider the nonholonomic problem of rolling without
slipping and twisting of a balanced ball over a fixed sphere in $\mathbb{R}^n$.
By relating the system to a modified LR system, we prove that the problem always has an invariant measure. Moreover,
this is a $SO(n)$-Chaplygin system that reduces to the cotangent bundle $T^*S^{n-1}$.
We present two integrable cases. The first one is obtained for a special inertia operator that allows the Chaplygin Hamiltonization of the reduced system.
In the second case, we consider the rigid body inertia operator
$\mathbb I\omega=I\omega+\omega I$, ${I=diag(I_1,\ldots,I_n)}$ with a symmetry $I_1=I_2=\ldots=I_{r} \ne I_{r+1}=I_{r+2}=\ldots=I_n$. It is shown that general trajectories
are quasi-periodic, while for $r\ne 1$, $n-1$ the Chaplygin reducing multiplier method does not apply.