2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/57555We explicitly demonstrate the existence of twistor and ambitwistor structure for the 4-dimensional complexified eikonal equation (CEE) and present its general solution consisting of two different classes. For both, every solution can be obtained from a generating twistor function in a purely algebraic way, via the procedure similar to that used in the Kerr theorem for shear-free null congruences. Bounded singularities of eikonal or of its gradient define some particle-like objects with nontrivial characteristics and dynamics. Example of a new static solution to CEE with a ring-like singularity is presented, and general principles of algebraic field theory - algebrodynamics - closely related to CEE are briefly discussed.6 pages, no figures. Proceedings of the V Asian-Pasific Conference on the Gravitation and Astrophysics, Moscow, September 2001Mathematical Physicstext