Quasi-Stäckel systems and two-dimensional Schrödinger equations in an electromagnetic field

We obtain the complete classification of two-dimensional Schrödinger equations in an electromagnetic field with an additional integral quadratic in momenta. For this, we use a Kovalevskaya-type change of variables and reduce the Hamiltonians to a quasi-Stäckel form. In that form, we perform the classification in the Painlevé sense and then return to the original variables.