We consider a class of billiard tables (X,g), where X is a smooth compact manifold of dimension two with smooth boundary \partial X and g is a smooth Riemannian metric on X, the billiard flow of which is completely integrable. The billiard table (X,g) is defined by means of a special double cover with two branched points and it admits a group of isometries G \cong \mathbb{Z}_2 \times\mathbb{Z}_2. Its boundary can be characterized by the string property; namely, the sum of distances from any point of \partial X to the branched points is constant. We provide examples of such billiard tables in the plane (elliptical regions), on the sphere \mathbf{S}^2,on the hyperbolic space \mathbf{H}^2, and on quadrics. The main result is that the spectrum of the corresponding Laplace–Beltrami operator with Robin boundary conditions involving a smooth function K on \partial X uniquely determines the function K, provided that K is invariant under the action of G.