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The goal of this thesis is the search for integrable and superintegrablesystems with magnetic field. We formulate the quantum mechanical determiningequations for second order integrals of motion in the cylindrical coordinatesand we find all quadratically integrable systems of the cylindrical type. Amongthem we search for systems admitting additional integrals of motion. We findall systems with an additional first order integral both in classical andquantum mechanics. It turns out that all these systems have already been knownand no other exist. We also find all systems with an additional integral oftype $L^2+\ldots$, respectively $L_y p_y-L_x p_y+\ldots$, of which the majorityis new to the literature. All found superintegrable systems admit the firstorder integral $L_z$ and we solve their Hamilton-Jacobi and Schr\"odingerequations by separation of variables in the cylindrical coordinates, for thefirst order systems in the Cartesian coordinates as well.

Integrable and superintegrable systems of cylindrical type in magnetic fields