## First integrals from conformal symmetries: Darboux-Koenigs metrics and beyond.(English)Zbl 1430.37063

Summary: On spaces of constant curvature, the geodesic equations automatically have higher order integrals, which are just built out of first order integrals, corresponding to the abundance of Killing vectors. This is no longer true for general conformally flat spaces, but in this case there is a large algebra of conformal symmetries. In this paper we use these conformal symmetries to build higher order integrals for the geodesic equations. We use this approach to give a new derivation of the Darboux-Koenigs metrics, which have only one Killing vector, but two quadratic integrals. We also consider the case of possessing one Killing vector and two cubic integrals. The approach allows the quantum analogue to be constructed in a simpler manner.

### MSC:

 37J37 Relations of finite-dimensional Hamiltonian and Lagrangian systems with Lie algebras and other algebraic structures 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 70H06 Completely integrable systems and methods of integration for problems in Hamiltonian and Lagrangian mechanics 70H20 Hamilton-Jacobi equations in mechanics 17B63 Poisson algebras 81S10 Geometry and quantization, symplectic methods
Full Text:

### References:

 [1] Dubrovin, B. A.; Fomenko, A. T.; Novikov, S. P., Modern Geometry - Methods and Applications (3 Volumes), (1984), Springer-Verlag: Springer-Verlag NY · Zbl 0529.53002 [2] Duval, C.; Valent, G., Quantum integrability of quadratic Killing tensors, J. Math. Phys., 46, Article 053516 pp., (2005), (22 pages) · Zbl 1110.81116 [3] Fordy, A. P., Quantum super-integrable systems as exactly solvable models, SIGMA, 3, 025, (2007), 10 pages · Zbl 1136.81027 [4] Fordy, A. P., Classical and quantum super-integrability: From Lissajous figures to exact solvability, Atom. Nuclei, 81, 832-842, (2018), preprint arXiv:1711.10583 [nlin.SI] [5] Fordy, A. P.; Huang, Q., Poisson algebras and 3D superintegrable Hamiltonian systems, SIGMA, 14, 022, (2018), 37 pages · Zbl 1416.17011 [6] Gilmore. Lie Groups, R., Lie Algebras and Some of their Applications, (1974), Wiley: Wiley New York [7] Kalnins, E. G.; Kress, J. M.; Miller Jr, W.; Winternitz, P., Superintegrable systems in Darboux spaces, J. Math. Phys., 44, 5811, (2003) · Zbl 1063.37050 [8] Kalnins, E. G.; Kress, J. M.; Winternitz, P., Superintegrability in a two-dimensional space of nonconstant curvature, J. Math. Phys., 43, 970, (2002) · Zbl 1059.37040 [9] Koenigs, G., Sur les géodésiques a intégrales quadratiques, (Darboux, G., Leçons sur la Théorie Générale des Surfaces, Vol. 4, (1972)), 368-404 [10] Matveev, V. S.; Shevchishin, V. V., Two-dimensional superintegrable metrics with one linear and one cubic integral, J. Geom. Phys., 61, 1353-1377, (2011) · Zbl 1218.53087 [11] Miller Jr, W.; Post, S.; Winternitz, P., Classical and quantum superintegrability with applications, J. Phys. A, 46, Article 423001 pp., (2013), (97 pages) · Zbl 1276.81070
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.