We consider Gevrey perturbations H of a completely integrable Gevrey Hamiltonian H 0 . Given a Cantor set Ω κ defined by a Diophantine condition, we find a family of KAM invariant tori of H with frequencies ω ∈ Ω κ which is Gevrey smooth in a Whitney sense. Moreover, we obtain a symplectic Gevrey normal form of the Hamiltonian in a neighborhood of the union Λ of the invariant tori. This leads to effective stability of the quasiperiodic motion near Λ. KAM theorem for Gevrey HamiltoniansLet D 0 be a bounded domain in R n , and T n = R n /2πZ n , n ≥ 2. We consider a class of real valued Gevrey Hamiltonians in T n × D 0 which are small perturbations of a real valued nondegenerate Gevrey Hamiltonian H 0 (I) depending only on the action variables I ∈ D 0 . Our aim is to obtain a family of KAM (Kolmogorov-Arnold-Moser) invariant tori Λ ω of H with frequencies ω in a suitable Cantor set Ω κ defined by a Diophantine condition and to prove Gevrey regularity for it. It turns out that for each ω ∈ Ω κ , Λ ω is a Gevrey smooth embedded torus having the same Gevrey regularity as the Hamiltonian H. Moreover, we shall prove that the family Λ ω , ω ∈ Ω κ , is Gevrey smooth with respect to ω in a Whitney sense, with a Gevrey index depending on the Gevrey class of H and on the exponent in the Diophantine condition. This naturally involves anisotropic Gevrey classes. Let ρ 1 , ρ 2 ≥ 1 and L 1 , L 2 be positive constants. Given a domain D ⊂ R n , we denote bywhere |α| = α 1 + · · · + α n and α! = α 1 ! · · · α n ! for α = (α 1 , . . . , α n ) ∈ N n . In the same way we define, and sometimes we do not indicate the Gevrey constants L 1 , L 2 .Let H 0 be a completely integrable real valued Gevrey smooth Hamiltonian T n × D 0 ∋ (θ, I) → H 0 (I) ∈ R. We suppose that H 0 is non-degenerate, which means that the map ∇H 0 : D 0 → Ω 0 is a diffeomorphism. Denote by g 0 ∈ C ∞ (Ω 0 ) the Legendre transform of H 0 (then ∇g 0 : Ω 0 → D 0 is the inverse map to ∇H 0 ). We suppose also that there are positive constants ρ > 1, A 0 > 0, and(Ω 0 ), andin the corresponding norms, defined as in (1.1). In particular, Ω 0 is a bounded domain. Given a subdomain D of D 0 we set Ω := ∇H 0 (D) ⊂ Ω 0 . Fix τ > n − 1 and κ > 0. We denote by Ω κ

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We consider a class of billiard tables (X, g), where X is a smooth compact manifold of dimension two with smooth boundary ∂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 ∼ = 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 2 , on the hyperbolic space 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 ∂X uniquely determines the function K, provided that K is invariant under the action of G.