Integrable Hamiltonian systems with vector potentials

We investigate integrable 2-dimensional Hamiltonian systems with scalar and vector potentials, admitting second invariants which are linear or quadratic in the momenta. In the case of a linear second invariant, we provide some examples of weakly-integrable systems. In the case of a quadratic second invariant, we recover the classical strongly-integrable systems in Cartesian and polar coordinates and provide some new examples of integrable systems in parabolic and elliptical coordinates.