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