We consider a billiard problem for compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. We show that there are two types of confocal families in such setting. Using an algebro-geometric integration technique, we prove that the billiard within generalized ellipses of each type is integrable in the sense of Liouville. Further, we prove a generalization of the Poncelet theorem and derive Cayley-type conditions for periodic trajectories and explore geometric consequences.

中文翻译：

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一张纸的双曲面共焦曲线内的伪欧几里得台球

我们考虑了一个由 Minkowski 空间中一张纸的双曲面上的共焦圆锥包围的紧凑域的台球问题。我们表明在这种情况下有两种类型的共焦家族。使用代数几何积分技术，我们证明了每种类型的广义椭圆内的台球在刘维尔意义上是可积分的。此外，我们证明了 Poncelet 定理的推广，并推导出周期性轨迹的 Cayley 类型条件并探索几何结果。