Magnetic Pole as Produced by a Point-like Electric Charge Embedded in Constant-Field Background

We consider a linear magnetic response to a point electric charge embedded in the background of parallel constant electric and magnetic fields in the framework of nonlinear electrodynamics. We find two types of responses. One is given by a vector potential free of any string singularity. The corresponding magnetic field may be thought of as two magnetic poles of opposite polarity coexisting at one point. The other response is given by a vector potential singular on a half-axis directed along the background fields. Its magnetic field is a magnetic monopole plus a field confined to an infinitely thin solenoid, whose role is the same as that of the Dirac string. The value of the magnetic charge is determined by the electric charge and the background fields and is expressed in terms of derivatives of the nonlinear local Lagrangian. Once the potential is singular, the nonlinear Maxwell equations written for potentials and field intensities are not equivalent. We argue why the preference should be given to potentials.