  In the present work we establish a simple relation between the Dirac equation
with a scalar and an electromagnetic potentials in a two-dimensional case and a
pair of decoupled Vekua equations. In general these Vekua equations are
bicomplex. However we show that the whole theory of pseudoanalytic functions
without modifications can be applied to these equations under a certain not
restrictive condition. As an example we formulate the similarity principle
which is the central reason why a pseudoanalytic function and as a consequence
a spinor field depending on two space variables share many of the properties of
analytic functions. One of the surprising consequences of the established
relation with pseudoanalytic functions consists in the following result.
Consider the Dirac equation with a scalar potential depending on one variable
with fixed energy and mass. In general this equation cannot be solved
explicitly even if one looks for wave functions of one variable. Nevertheless
for such Dirac equation we obtain an algorithmically simple procedure for
constructing in explicit form a complete system of exact solutions (depending
on two variables). These solutions generalize the system of powers of z in
complex analysis and are called formal powers. With their aid any regular
solution of the Dirac equation can be represented by its Taylor series in
formal powers.
