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 an invariant measure for these systems. 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 problem (with Vladimir Dragović and Borislav Gajić).