An Infinite Family of Maximally Superintegrable Systems in a Magnetic Field with Higher Order Integrals

TL;DR

This paper demonstrates that a class of three-dimensional superintegrable systems with magnetic fields are actually maximally superintegrable, with some integrals of motion being arbitrarily high order polynomials, expanding understanding of integrability in magnetic systems.

Contribution

The authors construct an additional integral of motion for a known class of superintegrable systems, proving their maximal superintegrability and revealing integrals of arbitrarily high polynomial order.

Findings

The system is maximally superintegrable.

New integrals can be of arbitrarily high polynomial order.

The class possesses periodic closed orbits.

Abstract

We construct an additional independent integral of motion for a class of three dimensional minimally superintegrable systems with constant magnetic field. This class was introduced in [J. Phys. A: Math. Theor. 50 (2017), 245202, 24 pages] and it is known to possess periodic closed orbits. In the present paper we demonstrate that it is maximally superintegrable. Depending on the values of the parameters of the system, the newly found integral can be of arbitrarily high polynomial order in momenta.