### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 037, 30 pages      arXiv:1810.13368      https://doi.org/10.3842/SIGMA.2019.037

### Generalised Darboux-Koenigs Metrics and 3-Dimensional Superintegrable Systems

Allan P. Fordy a and Qing Huang b
a) School of Mathematics, University of Leeds, Leeds LS2 9JT, UK
b) School of Mathematics, Northwest University, Xi'an 710069, People's Republic of China

Received November 01, 2018, in final form April 16, 2019; Published online May 05, 2019

Abstract
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.

Key words: Darboux-Koenigs metrics; Hamiltonian system; super-integrability; Poisson algebra; conformal algebra.

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