Integrable Schroedinger operators with magnetic fields

We consider the problem of classifying all pairs of commuting Schrodinger operators with magnetic terms in two degrees of freedom. We derive a concise set of necessary and sufficient conditions for two such operators to commute, and identify the difference compared with the corresponding conditions in the classical case. We classify all such pairs of operators in the case when one of them has a constant, non-zero magnetic field and a non-constant potential.