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

DOI10.1016/j.geomphys.2011.02.012zbMath1218.53087arXiv1010.4699OpenAlexW1998483550MaRDI QIDQ550282

Vladimir S. Matveev, Vsevolod V. Shevchishin

Publication date: 8 July 2011

Published in: Journal of Geometry and Physics (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/1010.4699

zbMATH Keywords

Killing vector fieldssuperintegrable systemspolynomially integrable geodesic flows on surfacessolvability of PDE

Mathematics Subject Classification ID

Related Items (27)

Integrable and superintegrable extensions of the rational Calogero–Moser model in three dimensions ⋮ Killing tensors on tori ⋮ Hexagonal Geodesic 3-Webs ⋮ Integrable cosmological potentials ⋮ Superintegrable systems on 3 dimensional conformally flat spaces ⋮ The gap phenomenon in the dimension study of finite type systems ⋮ Koenigs theorem and superintegrable Liouville metrics ⋮ Quasi-bi-Hamiltonian structures, complex functions and superintegrability: the Tremblay–Turbiner–Winternitz (TTW) and the Post–Winternitz (PW) systems ⋮ On the geometric and analytical properties of the anharmonic oscillator ⋮ Integrable geodesic flows and metrisable second-order ordinary differential equations ⋮ Superintegrable models on Riemannian surfaces of revolution with integrals of any integer degree. I ⋮ Superintegrable systems on 3-dimensional curved spaces: Eisenhart formalism and separability ⋮ Conformal Killing symmetric tensors on Lie groups ⋮ Integrability analysis of natural Hamiltonian systems in curved spaces ⋮ Zoll and Tannery metrics from a superintegrable geodesic flow ⋮ Superintegrable geodesic flows versus Zoll metrics ⋮ Open problems, questions and challenges in finite- dimensional integrable systems ⋮ Quadratic integrals of geodesic flow, webs, and integrable billiards ⋮ Explicit metrics for a class of two-dimensional cubically superintegrable systems ⋮ Hydrodynamic-type systems describing 2-dimensional polynomially integrable geodesic flows ⋮ Maximum rank of a Legendrian web ⋮ Global structure and geodesics for Koenigs superintegrable systems ⋮ Symmetric Killing tensors on nilmanifolds ⋮ First integrals from conformal symmetries: Darboux-Koenigs metrics and beyond ⋮ (Super-)integrable systems associated to 2-dimensional projective connections with one projective symmetry ⋮ Superintegrability on the hyperbolic plane with integrals of any degree \(\geq 2\) ⋮ The role of commuting operators in quantum superintegrable systems

Cites Work

• Unnamed Item
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• Two-dimensional metrics admitting precisely one projective vector field
• On a class of integrable systems with a cubic first integral
• Pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta, and proof of the projective Obata conjecture for two-dimensional pseudo-Riemannian metrics
• Invariant characterization of Liouville metrics and polynomial integrals
• Differential invariants for cubic integrals of geodesic flows on surfaces
• Compact Liouville surfaces
• New examples of integrable conservative systems on \(S^2\) and the case of Goryachev-Chaplygin
• A new integrable system on the sphere
• On integrals of the third degree in momenta
• Metrisability of two-dimensional projective structures
• Lichnerowicz-Obata conjecture in dimension two
• A solution of a problem of Sophus Lie: normal forms of two-dimensional metrics admitting two projective vector fields
• GEODESIC FLOWS ON TWO-DIMENSIONAL MANIFOLDS WITH AN ADDITIONAL FIRST INTEGRAL THAT IS POLYNOMIAL IN THE VELOCITIES
• Second-order superintegrable systems in conformally flat spaces. I. Two-dimensional classical structure theory
• Unified treatment and classification of superintegrable systems with integrals quadratic in momenta on a two-dimensional manifold
• Superintegrable systems with third-order integrals of motion
• Two-dimensional Riemannian metrics with integrable geodesic flows. Local and global geometry
• Superintegrable n=2 systems, quadratic constants of motion, and potentials of Drach
• Superintegrable systems in Darboux spaces
• Hamiltonians separable in Cartesian coordinates and third-order integrals of motion
• The Maupertuis principle and geodesic flows on the sphere arising from integrable cases in the dynamics of a rigid body
• Two-dimensional geodesic flows having first integrals of higher degree
• Quantum integrability of Beltrami-Laplace operator as geodesic equivalence

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