  We first construct nonholonomic systems of $n$ homogeneous balls $\mathbf
B_1,\dots,\mathbf B_n$ with centers $O_1,...,O_n$ and with the same radius $r$
that are rolling without slipping around a fixed sphere $\mathbf S_0$ with
center $O$ and radius $R$. In addition, it is assumed that a dynamically
nonsymmetric sphere $\mathbf S$ of radius $R+2r$ and the center that coincides
with the center $O$ of the fixed sphere $\mathbf S_0$ rolls without slipping
over the moving balls $\mathbf B_1,\dots,\mathbf B_n$. We prove that these
systems possess an invariant measure. As the second task, we consider the
limit, when the radius $R$ tends to infinity. We obtain a corresponding planar
problem consisting of $n$ homogeneous balls $\mathbf B_1,\dots,\mathbf B_n$
with centers $O_1,...,O_n$ and the same radius $r$ that are rolling without
slipping over a fixed plane $\Sigma_0$, and a moving plane $\Sigma$ that moves
without slipping over the homogeneous balls. We prove that this system
possesses an invariant measure and that it is integrable in quadratures
according to the Euler-Jacobi theorem.
