  We focus our attention, once again, on the Klein--Gordon and Dirac equations
with a plane-wave field. We recall that for the first time a set of solutions
of these equations was found by Volkov. The Volkov solutions are widely used in
calculations of quantum effects with electrons and other elementary particles
in laser beams. We demonstrate that one can construct sets of solutions which
differ from the Volkov solutions and which may be useful in physical
applications. For this purpose, we show that the transversal charge motion in a
plane wave can be mapped by a special transformation to transversal free
particle motion. This allows us to find new sets of solutions where the
transversal motion is characterized by quantum numbers different from Volkov's
(in the Volkov solutions this motion is characterized by the transversal
momentum). In particular, we construct solutions with semiclassical transversal
charge motion (transversal squeezed coherent states). In addition, we
demonstrate how the plane-wave field can be eliminated from the transversal
charge motion in a more complicated case of the so-called combined
electromagnetic field (a combination of a plane-wave field and constant
colinear electric and magnetic fields). Thus, we find new sets of solutions of
the Klein--Gordon and Dirac equations with the combined electromagnetic field.
