New classes of quadratically integrable systems in magnetic fields: The generalized cylindrical and spherical cases

We study integrable and superintegrable systems with magnetic field possessing quadratic integrals of motion on the three-dimensional Euclidean space. In contrast case without vector potential, corresponding may no longer be connected to separation variables in Hamilton-Jacobi equation can have more general leading order terms. focus two cases extending physically relevant cylindrical- spherical-type integrals. find three new generalized cylindrical but none spherical one. conjecture that this is related presence, respectively absence, maximal abelian Lie subalgebra algebra generated by first limit vanishing field. By investigating superintegrability, we only one (minimally) system among ones. It does not separate any orthogonal coordinate system. This provides a mathematical model helical undulator placed an infinite solenoid.