This paper proves two main results. First, it is shown that if Σ is a smooth manifold homeomorphic to the standard n-torus Tn = Rn / Zn and H is a real-analytically completely integrable convex hamiltonian on T* Σ, then Σ is diffeomorphic to Tn. Second, it is proven that for some topological 7-manifolds, the cotangent bundle of each smooth structure admits a real-analytically completely integrable riemannian metric hamiltonian.