  We present the classification of quadratically integrable systems of the
cylindrical type with magnetic fields in quantum mechanics. Following the
direct method used in classical mechanics by [F Fournier et al 2020 J. Phys. A:
Math. Theor. 53 085203] to facilitate the comparison, the cases which may a
priori differ yield 2 systems without any correction and 2 with it. In all of
them, the magnetic field $B$ coincides with the classical one, only the scalar
potential $W$ may contain a $\hbar^2$-dependent correction. Two of the systems
have both cylindrical integrals quadratic in momenta and are therefore not
separable. These results form a basis for a prospective study of
superintegrability.
