Family of nonstandard integrable and superintegrable classical Hamiltonian systems in non-vanishing magnetic fields

TL;DR

This paper constructs a comprehensive family of nonstandard integrable and superintegrable classical Hamiltonian systems in magnetic fields, generalizing previous specific examples and identifying conditions for superintegrability.

Contribution

It introduces a broad family of nonstandard integrable systems in magnetic fields, extending known cases and exploring superintegrability conditions.

Findings

Family of systems with multiple parameters constructed

Systems reduce to known integrable cases in certain limits

Superintegrability achieved with additional integrals

Abstract

In this paper we present the construction of all nonstandard integrable systems in magnetic fields whose integrals have leading order structure corresponding to the case (i) of Theorem 1 in [A Marchesiello and L \v{S}nobl 2022 {\it J. Phys. A: Math. Theor.} {\bf 55} 145203]. We find that the resulting systems can be written as one family with several parameters. For certain limits of these parameters the system belongs to intersections with already known standard systems separating in Cartesian and / or cylindrical coordinates and the number of independent integrals of motion increases, thus the system becomes minimally superintegrable. These results generalize the particular example presented in section 3 of [A Marchesiello and L \v{S}nobl 2022 {\it J. Phys. A: Math. Theor.} {\bf 55} 145203].