The Fock-Darwin system is analysed from the point of view of its symmetry properties in the
quantum and classical frameworks. The quantum Fock-Darwin system is known to have two
sets of ladder operators, a fact which guarantees its solvability. We show that for rational
values of the quotient of two relevant frequencies, this system is superintegrable, the quantum
symmetries being responsible for the degeneracy of the energy levels. These symmetries are
of higher order and close a polynomial algebra. In the classical case, the ladder operators are
replaced by ladder functions and the symmetries by constants of motion. We also prove that
the rational classical system is superintegrable and its trajectories are closed. The constants
of motion are also generators of symmetry transformations in the phase space that have been
integrated for some special cases. These transformations connect different trajectories with
the same energy. The coherent states of the quantum superintegrable system are found and
they reproduce the closed trajectories of the classical one.
