  The Abelian Born-Infeld classical non-linear electrodynamic has been used to
investigate the electric and magnetostatic fields generated by a point-like
electrical charge at rest in an inertial frame. The results show a rich
internal structure for the charge. Analytical solutions have also been found.
Such findings have been interpreted in terms of vacuum polarization and
magnetic-like charges produced by the very high strengths of the electric field
considered. Apparently non-linearity is to be accounted for the emergence of an
anomalous magnetostatic field suggesting a possible connection to that created
by a magnetic dipole composed of two mognetic charges with opposite signals.
Consistently in situations where the Born-Infeld field strength parameter is
free to become infinite, Maxwell`s regime takes over, the magnetic sector
vanishes and the electric field assumes a Coulomb behavior with no trace of a
magnetic component. The connection to other monopole solutions, like Dirac`s,
t' Hooft`s or Poliakov`s types, are also discussed. Finally some speculative
remarks are presented in an attempt to explain such fields.
