  A minimal truncated set of the integral Dyson-Schwinger equations, in
Minkowski spacetime, that allows to explore QED beyond its perturbative
solution is derived for general linear covariant gauges. The minimal set
includes the equations for the fermion and photon propagators, the
photon-fermion vertex, and the two-photon-two-fermion one-particle-irreducible
diagram. If the first three equations are exact, to build a closed set of
equations, the two-photon-two-fermion equation is truncated ignoring the
contribution of Green functions with large number of external legs. It is shown
that the truncated equation for the two-photon-two-fermion vertex reproduces
the lowest-order perturbative result in the limit of the small coupling
constant. Furthermore, this equation allows to define an iterative procedure to
compute higher order corrections in the coupling constant. The Ward-Takahashi
identity for the two-photon-two-fermion irreducible vertex is derived and
solved in the soft photon limit, where one of the photon momenta vanish, in the
low photon momenta limit and for general kinematics. The solution of the
Ward-Takahashi identity determines the longitudinal component of the
two-photon-two-fermion irreducible vertex, while it is proposed to use the
Dyson-Schwinger equation to determine the transverse part of this irreducible
diagram. The two-photon-two-fermion DSE is solved in heavy fermion limit,
considering a simplified version of the QED vertices. The contribution of this
irreducible vertex to a low-energy effective photon-fermion vertex is discussed
and the fermionic operators that are generated are computed in terms of the
fermion propagator functions.
