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

Cylindrically symmetric quantum mechanical systems with position dependent masses (PDM) 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 twenty seven superintegrable and twelve maximally superintegrable. The arbitrary elements of the correspondinding Hamiltonians (i.e.,masses and potentials) are presented explicitly.