We consider a class of billiard tables (X; g), where X is a smooth compact manifold of dimension 2 with smooth boundary @X and g is a smooth Riemannian metric on X , the billiard ow of which is completely integrable. The billiard table (X; g) is de ned by means of a special double cover with two branched points and it admits a group of isometries G = Z 2 Z 2 . Its boundary can be characterized by the string property, namely, the sum of distances from any point of @X to the branched points is constant. We provide examples of such billiard tables in the plane (elliptical regions), on the sphere S , on the hyperbolic space , and on quadrics. The main result is that the spectrum of the corresponding LaplaceBeltrami operator with Robin boundary conditions involving a smooth function K on @X determines uniquely the function K provided that K is invariant under the action of G .

We consider integrable system on the sphere S 2 with an additional integral of fourth order in the momenta. At the special values of parameters this system coincides with the Kowalevski-Goryachev-Chaplygin system. 1