Single- and multi-valued solutions of homogeneous Maxwell equations in vacuum
are considered, with ''sources'' formed by the (point- or string-like)
singularities of the field strengths and, generally, irreducible to any
delta-functions' distribution. Maxwell equations themselves are treated as
consequences (say, integrability conditions) of a primary ``superpotential''
field subject to some nonlinear and over-determined constraints (related, in
particular, to twistor structures). As the result, we obtain (in explicit or
implicit algebraic form) a distinguished class of Maxwell fields, with singular
sources necessarily carrying a ``self-quantized'' electric charge integer
multiple to a minimal ``elementary'' one. Particle-like singular objects are
subject to the dynamics consistent with homogeneous Maxwell equations and
undergo transmutations -- bifurcations of different types. The presented scheme
originates from the ``algebrodynamical'' approach developed by the author and
reviewed in the last section. Incidentally, fundamental equivalence relations
between the solutions of Maxwell equations, complex self-dual conditions and of
Weyl ``neutrino'' equations are established, and the problem of magnetic
monopole is briefly discussed.