We investigate integrable two-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