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

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 the 2-sphere. 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

💡 Research Summary

The paper undertakes a complete local classification of two‑dimensional Riemannian metrics whose geodesic flows admit two independent first integrals: one linear in the momenta (hence generated by a Killing vector) and one cubic in the momenta. Starting from the existence of a linear integral, the authors bring the metric into a normal form (g=\lambda(x),dx^{2}+\mu(x),dy^{2}) by choosing coordinates adapted to the Killing field. The condition that a homogeneous cubic polynomial (F) in the momenta Poisson‑commutes with the Hamiltonian (H=\frac12 g^{ij}p_{i}p_{j}) yields a coupled system of nonlinear differential equations for (\lambda,\mu) and the coefficients of (F). By systematic elimination the authors reduce this system to a single third‑order nonlinear ordinary differential equation for a single function (essentially the conformal factor of the metric).

The ODE admits two trivial families corresponding to constant curvature spaces (the sphere, the Euclidean plane, and the hyperbolic plane). In addition, a rich non‑trivial family is obtained, parametrised by a real constant (c) and, in some formulations, by an arbitrary smooth function. A representative metric from this family is
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