  We derive a set of coupled four-dimensional integral equations for the
$NN-\pi NN$ system using our modified version of the Taylor method of
classification-of-diagrams. These equations are covariant, obey two and
three-body unitarity and contain subtraction terms which eliminate the
double-counting present in some previous four-dimensional $NN-\pi NN$
equations. The equations are then recast into a from convenient for computation
by grouping the subtraction terms together and obtaining a set of two-fragment
scattering equations for the amplitudes of interest.
