In this paper we study the motion of a charged particle in the presence of a magnetic field
created by three different systems of wires: an infinite rectilinear filament, a circular wire and
the union of both. In the first case we prove that the equations of motion are Liouville
integrable and we provide a complete description of the trajectories, which turn out to be of
helicoidal type. In the case of the circular wire we study some restricted motions and we
show that there is a trapping region similar to the Van Allen inner radiation belt in the Earth …