  Integrable quantum mechanical systems with magnetic fields are constructed in
two-dimensional Euclidean space. The integral of motion is assumed to be a
first or second order Hermitian operator. Contrary to the case of purely scalar
potentials, quadratic integrability does not imply separation of variables in
the Schroedinger equation. Moreover, quantum and classical integrable systems
do not necessarily coincide: the Hamiltonian can depend on the Planck constant
in a nontrivial manner.
