  This paper continues the author's work \cite{PartI}, where a new framework of
the matter-induced physical geometry was built and an intrinsic nonlinearity of
the Dirac equation discovered. Here, the nonlinear Dirac equation is solved and
the localized configurations are found analytically. Of the two possible types
of the potentially stationary localized configurations of the Dirac field, only
one is stable with respect to the action of an external field and it
corresponds to a positive charge. A connection with the global charge asymmetry
of matter in the Universe and with the recently observed excess of the cosmic
positrons is discussed.
