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

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 Marchesiello and Snobl (2022 J. Phys. A: Math. Theor. 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 Marchesiello and Snobl (2022 J. Phys. A: Math. Theor. 55 145203).

Automated summary

In this paper, the construction of all nonstandard integrable systems in magnetic fields is presented. These systems have integrals with leading order structure corresponding to the case (i) of theorem 1 in Marchesiello and Snobl (2022 J. Phys. A: Math. Theor. 55 145203). The resulting systems can be written as one family with several parameters. For certain limits of these parameters, the system becomes minimally superintegrable by belonging to intersections with already known standard systems in Cartesian and/or cylindrical coordinates, and the number of independent integrals of motion increases. These results generalize a particular example presented in section 3 of Marchesiello and Snobl (2022 J. Phys. A: Math. Theor. 55 145203).