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 with the moving balls \(\mathbf{B}_{1},\dots,\mathbf{B}_{n}\). The problem is considered in four different configurations, three of which are new. We derive the equations of motion and find an invariant measure for these systems. As the main result, for \(n=1\) we find two cases that are integrable by 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, which roll without slipping over a fixed plane \(\Sigma_{0}\) with a plane \(\Sigma\) that moves without slipping over these balls.

我们考虑具有相同半径\ (r\) 的\(n\)个均匀球\(\mathbf{B}_{1},\dots,\mathbf{B}_{n}\) 的非完整系统围绕中心\(O\)和半径\(R\) 的固定球体\(\mathbf{S}_{0}\)滚动而不滑动。此外，假设一个动态非对称球体\(\mathbf{S}\)的中心与固定球体\(\mathbf{S}_{0}\ ) 的中心 \(O\ ) 重合滚动而不打滑接触移动的球\(\mathbf{B}_{1},\dots,\mathbf{B}_{n}\). 该问题在四种不同的配置中被考虑，其中三种是新的。我们推导出运动方程并找到这些系统的不变度量。作为主要结果，对于\(n=1\)，我们发现根据欧拉 - 雅可比定理可通过正交积分的两种情况。获得的可积非完整模型是著名的 Chaplygin 球可积问题的自然扩展。此外，我们明确地整合了由\(n\)个半径相同但质量不同的均匀球组成的平面问题，这些球在固定平面\(\Sigma_{0}\)上滚动而不滑动，平面\(\ Sigma\)移动而不会滑过这些球。