Integrability of magnetic fields created by current distributions

The existence of first integrals and periodic orbits of magnetic fields created by thin wires is investigated. When the current lines are planar we prove that magnetic orbits are closed near the wires and we provide two examples of magnetic fields without polynomial first integrals, thus contradicting Stefanescu's conjecture. When the current lines are non-planar we provide some examples of rectilinear configurations giving rise to helicoidal orbits near the wires and to chaotic portraits.