Topological classification of integrable geodesic flows on orientable two-dimensional Riemannian manifolds with additional integral depending on momenta linearly or quadratically (Q1326930)

Kolokotsov showed that the geodesic flow on the 2-sphere \(S^ 2\) or the 2-torus \(T^ 2\) equipped with some Riemannian metric has an additional quadratic integral functionally independent of the energy integral if and only if the metric can be written in a special form (a Liouville metric in the case of \(T^ 2\)). To such metrics one can associate certain combinatorial invariants called the code of the metric. In the paper the classification problem modulo topological equivalence for the geodesic flow on such a surface is related to the classification of the codes of the metric.