  The method of solving the Bethe-Salpeter equation in Minkowski space, which
we developed previously for spinless particles, is extended to a system of two
fermions. The method is based on the Nakanishi integral representation of the
amplitude and on projecting the equation on the light-front plane. The
singularities in the projected two-fermion kernel are regularized without
modifying the original Bethe-Salpeter amplitudes. The numerical solutions for
the J=0 bound state with the scalar, pseudoscalar and massless vector exchange
kernels are found. The stability of the scalar and positronium states without
vertex form factor is discussed. Binding energies are in close agreement with
the Euclidean results. Corresponding amplitudes in Minkowski space are
obtained.
