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Two-dimensional superintegrable metrics with one linear and one cubic integral. (English) Zbl 1218.53087

Summary: 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 \(S^{2}\). 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.

MSC:

53D25 Geodesic flows in symplectic geometry and contact geometry
53B20 Local Riemannian geometry
53B21 Methods of local Riemannian geometry
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53A55 Differential invariants (local theory), geometric objects
53A35 Non-Euclidean differential geometry
37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
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