  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.
