On the nonintegrability of magnetic field lines

Abstract

Agraïments: The second author is partially supported by FCT through CAMGDS, Lisbon.

We show that a magnetic field created by a simple planar configuration of three rectilinear wires may not be holomorphically integrable when considered as a vector field in C3. 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. This is also an alternative way of contradicting the Stefanescu conjecture in the polynomial setting.