  We study the Maxwell-Dirac equations in a manifestly gauge invariant
presentation using only the spinor bilinear scalar and pseudoscalar densities,
and the vector and pseudovector currents, together with their quadratic Fierz
relations. The internally produced vector potential is expressed via algebraic
manipulation of the Dirac equation, as a rational function of the Fierz
bilinears and first derivatives (valid on the support of the scalar density),
which allows a gauge invariant vector potential to be defined. This leads to a
Fierz bilinear formulation of the Maxwell tensor and of the Maxwell-Dirac
equations, without any reference to gauge dependent quantities. We show how
demanding invariance of tensor fields under the action of a fixed (but
arbitrary) Lie subgroup of the Poincar\'e group leads to symmetry reduced
equations. The procedure is illustrated, and the reduced equations worked out
explicitly for standard spherical and cylindrical cases, which are coupled
third order nonlinear PDEs. Spherical symmetry necessitates the existence of
magnetic monopoles, which do not affect the coupled Maxwell-Dirac system due to
magnetic terms canceling. In this paper we do not take up numerical
computations. As a demonstration of the power of our approach, we also work out
the symmetry reduced equations for two distinct classes of dimension 4
one-parameter families of Poincar\'e subgroups, one splitting and one
non-splitting. The splitting class yields no solutions, whereas for the
non-splitting class we find a family of formal exact solutions in closed form.
