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Any smooth geodesic flow is locally integrable with smooth integrals. We showthat generically this fails if we require, in addition, that the integrals arepolynomial (or, more generally, analytic) in momenta. Consequently we obtainthat a generic real-analytic metric does not admit, even locally, areal-analytic integral.

The geodesic flow of a generic metric does not admit nontrivial integrals polynomial in momenta