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This paper studies smooth obstructions to integrability and proves two mainresults. First, it is shown that if a smooth topological n-torus admits areal-analytically completely integrable convex hamiltonian on its cotangentbundle, then the torus is diffeomorphic to the standard n-torus. This is thefirst known result where the smooth structure of a manifold obstructs completeintegrability. Second, it is proven that each one of the Witten-Kreck-Stolz7-manifolds admit a real-analytically completely integrable geodesic flow onits cotangent bundle. This gives examples of topological manifolds all of whosesmooth structures admit a real-analytically completely integrable convexhamiltonian on its cotangent bundle. Additional examples are provided byEschenburgh and Aloff-Wallach spaces.

The Maslov cocycle, smooth structures and real-analytic complete integrability