Integrable billiards on a Minkowski hyperboloid: extremal polynomials and topology

We consider billiard systems within compact domains bounded by confocal conics on a hyperboloid of one sheet in the Minkowski space. derive conditions for elliptic periodicity such billiards. describe topology those terms Fomenko invariants. provide then functional Pell equations and related extremal polynomials. Several examples are computed functions classical Chebyshev Zolotarev polynomials, as polynomials over or two intervals. These results contrasted with cases billiards Euclidean planes.