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