Field-particle structures explicitly defined by any shear-free null congruence

Celebrated Kerr theorem pus in correspondence twistor functions in $CP^3$ and shear-free null congruences in M. We demonstrate that any twistor function algebraically defines also a whole set of relativistic vacuum fields – Maxwell, Yang-Mills, Weyl and eikonal ones, together with a Kerr-Shild metric. Common singularities of these fields possess some properties of quantum particles (e.g., the integer-valued electric charges) and are involved in a nontrivial “algebraic” dynamics. The latter turns out to be conservative for a wide class of holomorphic generating twistor funcions.