In this article, we consider mechanical billiard systems defined with Lagrange’s integrable extension of Euler’s two-center problems in the Euclidean space, the sphere, and the hyperbolic space of arbitrary dimension \(n\geqslant 3\). In the three-dimensional Euclidean space, we show that the billiard systems with any finite combinations of spheroids and circular hyperboloids of two sheets having two foci at the Kepler centers are integrable. The same holds for the projections of these systems on the three-dimensional sphere and in the three-dimensional hyperbolic space by means of central projection. Using the same approach, we also extend these results to the \(n\)-dimensional cases.

在本文中，我们考虑用欧拉空间、球体和任意维度\(n\geqslant 3\)的双曲空间中欧拉二中心问题的拉格朗日可积扩展定义的机械台球系统。在三维欧几里得空间中，我们证明了具有两个焦点在开普勒中心的两片的椭球体和圆形双曲面的任意有限组合的台球系统是可积的。这同样适用于这些系统通过中心投影在三维球体和三维双曲空间中的投影。使用相同的方法，我们还将这些结果扩展到\(n\)维情况。