Some properties of the magnetic fields generated by symmetric configurations of wires

In this paper we study some properties of the magnetic field lines and their effect on the particle motions. For certain configurations of wires we prove the existence of first integrals and ergodic (quasi-periodic) orbits. When the magnetic fields possess a Euclidean symmetry we prove that the equations of motion (in the relativistic and non-relativistic cases) inherit a first integral different from the kinetic energy. As a consequence of this property we show that in some physical examples the wires creating the magnetic field are unreachable for electric charges and there exist confinement spatial regions. Part of these mathematical results are of interest to electrical engineers, helping to keep the power lines electrically neutral.