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Weak integrability of Hamiltonians on the two torus and rigidity

We show that a C∞ k-basic Finsler metric on the two torus T2 whose geodesic flow preserves a codimension one C1,L foliation is in fact flat. Although integrable high energy levels of Hamiltonians on the torus are not flat in general, the C1,L integrability of k-basic Finsler geodesic flows on T2 implies flatness and in particular, C∞ integrability. We also show that a codimension one C1 foliation invariant by the geodesic flow of any Finsler metric on T2 must coincide with the Busemann foliation. A consequence of the above results is that the Hopf conjecture would be false for k-basic Finsler metrics on the two torus if and only if there exists a C0 integrable k-basic Finsler geodesic flow that is not C1,L integrable.