Abstract: The talk presents families of integrable and superintegrable classical Hamiltonian systems in magnetic fields. We consider more general structure of their quadratic commuting integrals of motion whose leading order terms are elements of the universal enveloping algebra of the three--dimensional Euclidean algebra. We show how these pairs of commuting elements lead to distinct independent integrals of motion in several nonvanishing magnetic fields. We also search for additional first-- and second--order integrals of motion of these systems to arrive at superintegrable systems. We construct the corresponding Poisson algebras of integrals of motion.

The talk is based on joint work with Libor \v{S}nobl.