On the nonintegrability of magnetic field lines

We prove the existence of a magnetic field created by a planar configuration of piecewise rectilinear wires which is not holomorphically integrable when considered as a vector field in C 3 . This is a counterexample to the S. Stefanescu conjecture (1986) in the holomorphic setting. In particular the method of the proof gives an easy way of showing that the corresponding real vector field does not admit a real polynomial first integral which provides also an alternative way of contradicting the Stefanescu conjecture in the polynomial setting.