We consider the nonholonomic systems of $n$ homogeneous balls $\mathbf
B_1,\dots,\mathbf B_n$ with the same radius $r$ that are rolling without
slipping about a fixed sphere $\mathbf S_0$ with center $O$ and radius $R$. In
addition, it is assumed that a dynamically nonsymmetric sphere $\mathbf S$ with
the center that coincides with the center $O$ of the fixed sphere $\mathbf S_0$
rolls without slipping in contact to the moving balls $\mathbf
B_1,\dots,\mathbf B_n$. The problem is considered in four different
configurations. We derive the equations of motion and prove that these systems
possess an invariant measure. As the main result, for $n=1$ we found two cases
that are integrable in quadratures according to the Euler-Jacobi theorem. The
obtained integrable nonholonomic models are natural extensions of the
well-known Chaplygin ball integrable problems. Further, we explicitly integrate
the planar problem consisting of $n$ homogeneous balls of the same radius, but
with different masses, that roll without slipping over a fixed plane $\Sigma_0$
with a plane $\Sigma$ that moves without slipping over these balls.
