Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2019
• Main Authors: Fordy, A.P., Huang, Q.
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2019
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/210185
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems / A.P. Fordy, Q. Huang // Symmetry, Integrability and Geometry: Methods and Applications. — 2019. — Т. 15. — Бібліогр.: 15 назв. — англ.

The Darboux-Koenigs metrics in 2D are an important class of conformally flat, non-constant curvature metrics with a single Killing vector and a pair of quadratic Killing tensors. In [arXiv:1804.06904], it was shown how to derive these by using the conformal symmetries of the 2D Euclidean metric. In this paper, we consider the conformal symmetries of the 3D Euclidean metric and similarly derive a large family of conformally flat metrics possessing between 1 and 3 Killing vectors (and therefore not constant curvature), together with a number of quadratic Killing tensors. We refer to these as generalised Darboux-Koenigs metrics. We thus construct multi-parameter families of super-integrable systems in 3 degrees of freedom. Restricting the parameters increases the isometry algebra, which enables us to fully determine the Poisson algebra of first integrals. This larger algebra of isometries is then used to reduce from 3 to 2 degrees of freedom, obtaining Darboux-Koenigs kinetic energies with potential functions, which are specific cases of the known super-integrable potentials.