Abstract
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^2ℏ2-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.
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