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

Abstract

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

35 pages

References in corpus (4)

• Superintegrable Systems in Darboux spaces
• Hamiltonians separable in cartesian coordinates and third-order integrals of motion
• Invariant characterization of Liouville metrics and polynomial integrals
• Superintegrable Systems with a Third Order Integrals of Motion

Cited by in corpus (28)

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• On natural Poisson bivectors on the sphere
• Explicit metrics for a class of two-dimensional superintegrable systems
• Higher-order superintegrability of a Holt related potential
• Superintegrable Systems on 3 Dimensional Conformally Flat Spaces
• Eisenhart Lift of $2$--Dimensional Mechanics
• First Integrals from Conformal Symmetries: Darboux-Koenigs Metrics and Beyond
• Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems
• Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows
• Symmetric Killing tensors on nilmanifolds
• Integrable Cosmological Potentials
• Killing tensors on tori
• Zoll and Tannery metrics from a superintegrable geodesic flow
• Quadratic integrals of geodesic flow, webs, and integrable billiards
• (Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry
• Global structure and geodesics for Koenigs superintegrable systems
• Adding Potentials to Superintegrable Systems with Symmetry
• Integrable and Superintegrable Extensions of the Rational Calogero-Moser Model in 3 Dimensions
• Koenigs Theorem and Superintegrable Liouville Metrics
• The gap phenomenon in the dimension study of finite type systems
• Superintegrable systems with position dependent mass: master symmetry and action-angle methods
• Local high-degree polynomial integrals of geodesic flows and the generalized hodograph method
• Hexagonal geodesic 3-webs
• Maximum rank of a Legendrian web
• Superintegrable metrics on surfaces admitting integrals of degrees 1 and 4