On superintegrability of 3D axially-symmetric non-subgroup-type systems with magnetic fields

Abstract

We extend the investigation of three-dimensional (3D) Hamiltonian systems of non-subgroup type admitting non-zero magnetic fields and an axial symmetry. Three different integrable cases are considered as starting points for superintegrability: the circular parabolic case, the oblate spheroidal case and the prolate spheroidal case. These integrable cases were introduced in [1]. In this paper, we focus on linear and some special cases of quadratic superintegrability. We investigated all possible additional linear integrals of motion for the oblate and the prolate spheroidal cases, separately. For both cases, no previously unknown superintegrable system arises. In addition, we looked for special quadratic integrals of motion for all three cases. We found one new minimally superintegrable system that lies at the intersection of the circular parabolic and cylindrical cases. We also found one new minimally superintegrable system that lies at the intersection of the cylindrical, spherical, oblate spheroidal and prolate spheroidal cases. By imposing additional conditions on these systems, we found for each quadratically minimally superintegrable system a new infinite family of higher-order maximally superintegrable systems linked with the caged and harmonic oscillator without magnetic fields, respectively, through a time-dependent canonical transformation.Comment: 12 figures, preprin