  A reduction of the Dirac-Maxwell equations in the case of static cylindrical
symmetry is performed. The behaviour of the resulting system of o.d.e.'s is
examined analytically and numerical solutions presented. There are two classes
of solutions.
  The first type of solution is a Dirac field surrounding a charged "wire". The
Dirac field is highly localised, concentrated in cylindrical shells about the
wire. A comparison with the usual linearized theory demonstrates that this
localization is entirely due to the non-linearities in the equations which
result from the inclusion of the "self-field".
  The second class of solutions have the electrostatic potential finite along
the axis of symmetry but unbounded at large distances from the axis.
