Perturbations of the hydrogen atom by inhomogeneous static electric and magnetic fields

The hydrogen atom is a specific, and physically relevant, Keplerian two-body system with one of the bodies assumed infinitely heavy. Isochronous integrable n-shell approximations for the perturbations of the hydrogen atom, known as Stark and Zeeman effects in the presence of homogeneous electric and magnetic fields, respectively, have been widely studied. In order to go beyond the set of linear systems, we considered perturbations by inhomogeneous electromagnetic fields. In particular, we found that the perturbation by a generic inhomogeneous magnetic field results in an approximately integrable anisochronous system which can be described as two rigid Euler tops coupled through higher order terms.