  The four-dimensional $NN-\pi NN$ equations are adapted to the case of scalar
particles with a $\phi^2\sigma$ interaction Lagrangian, and solved for energies
below the $\sigma$-production threshold. This is achieved in the approximation
where $\phi\sigma$ scattering is dominated by the $s$-channel $\phi$-pole term.
The importance of the removal of double-counting is investigated, and a
detailed comparison of the results of a covariant coupled-channels formulation
and the Bethe-Salpeter equation in the ladder and ladder plus crossed-box
approximations is presented. A brief discussion of the extension of the method
to energies above the $\sigma$-production threshold is given.
