  When exploring equations of nonlinear electrodynamics in effective medium
formed by mutually parallel external electric and magnetic fields, we come to
special static axial-symmetric solutions of two types. The first are comprised
of fields referred to as electric and magnetic responses to a point-like
electric charge when placed into the medium. In electric case, this is a field
determined by the induced charge density. In magnetic case, this is a field
carrying no magnetic charge and determined by an induced current. Fields of
second type require presence of pseudoscalar constants for their existence.
These are singular on the axis drawn along the external fields. In electric
case this is a field of an inhomogeneously charged infinitely thin thread. In
magnetic case this is the magnetic monopole with the Dirac string supported by
solenoidal current. In both cases the necessary pseudoscalar constant is
supplied by field derivatives of nonlinear Lagrangian taken on external fields.
There is also a magnetic thread solution dual to electric thread with null
total magnetic charge.
