Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 1. Systems with cylindric symmetry

Abstract Cylindrically symmetric quantum mechanical systems with position dependent masses admitting at least one second order integral of motion are classified. It is proved that there exist 68 such systems which are inequivalent. Among them there are thirty superintegrable and twelve maximally superintegrable ones. The arbitrary elements of the corresponding Hamiltonians (i.e.,masses and potentials) are presented explicitly.