  We give a brief overview of a non-Lagrangian approach to field theory based
on a generalization of the Kerr-Penrose theorem and algebraic twistor
equations. Explicit algorithms for obtaining the set of fundamental (Maxwell,
SL(2, C)-Yang-Mills, spinor Weyl and curvature) fields associated with every
solution of the basic system of algebraic equations are reviewed. The notion of
a boost-invariant solution is introduced, and the unique axially-symmetric and
boost-invariant solution which can be generated by twistor functions is
obtained, together with the associated fields. It is found that this solution
possesses a wide variety of point-, string- and membrane-like singularities
exhibiting nontrivial dynamics and transmutations.
