CLOSED GEODESICS AND BILLIARDS ON QUADRICS RELATED TO ELLIPTIC KdV SOLUTIONS

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 in ${mathbb R}^n$. 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.