  We consider the distribution of the axionic dark matter and the profiles of
the axionically induced electric field on the background of a regular magnetic
monopole. The background static spherically symmetric spacetime is considered
to be regular at the center of the object and has no event horizons, the
nonminimal interaction of strong gravitational and monopole type magnetic
fields being the source, which provides this regularity. We assume that the
pseudoscalar (axion) field also is nonminimally coupled to the electromagnetic
and gravitational fields, and we search for solutions to the extended equations
of nonminimal axion electrostatics, which satisfy two requirements: first, the
axionically induced radial electric field vanishes at the center; second, the
pseudoscalar field is regular at the center of the formed axionic dyon. We have
found the set of parameters, describing the nonminimal axion-photon coupling,
for which both requirements are satisfied. Properties of the two-parameter
family of regular solutions to the equations of nonminimal axion electrostatics
are analyzed numerically. The regular axionically induced electric field is
shown to have the structure, which is typical for a simple electric layer.
