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Статья

Two-dimensional superintegrable metrics with one linear and one cubic integral

Journal of Geometry and Physics. 2011. Vol. 61. No. 8. P. 1353-1377.
Matveev V. S., Shevchishin V.

We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta. We also show that some of these metrics can be extended to S2. This gives us new examples of Hamiltonian systems on the sphere with integrals of degree three in momenta, and the first examples of superintegrable metrics of nonconstant curvature on a closed surface.