  We consider algebraic geometrical properties of the integrable billiard on a
quadric Q with elastic impacts along another quadric confocal to Q. These
properties are in sharp contrast with those of the ellipsoidal Birkhoff
billiards. Namely, generic complex invariant manifolds are not Abelian
varieties, and the billiard map is no more algebraic. A Poncelet-like theorem
for such system is known. We give explicit sufficient conditions both for
closed geodesics and periodic billiard orbits on Q and discuss their relation
with the elliptic KdV solutions and elliptic Calogero system
