  The criterion of differentiability of functions of quaternion variable is
used as the basis of some algebraic field theory. Its necessary consequences
are free Maxwell and Yang-Mills equations. The differentiability equations may
be integrated in twistor variables and are reduced to algebraic ones. In the
article we present bisingular solution and its topological modifications.
Related EM-fields appear to be just the well-known Born solution and its
modifications with singular structure of ring-like and toroidal topology.
General problems of the algebrodynamical approach are discussed.
