  We 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.
