  The two-fermion relativistic wave equations of Constraint Theory are reduced,
after expressing the components of the $4\times 4$ matrix wave function in
terms of one of the $2\times 2$ components, to a single equation of the
Pauli-Schr\"odinger type, valid for all sectors of quantum numbers. The
potentials that are present belong to the general classes of scalar,
pseudoscalar and vector interactions and are calculable in perturbation theory
from Feynman diagrams. In the limit when one of the masses becomes infinite,
the equation reduces to the two-component form of the one-particle Dirac
equation with external static potentials. The Hamiltonian, to order $1/c^2$,
reproduces most of the known theoretical results obtained by other methods. The
gauge invariance of the wave equation is checked, to that order, in the case of
QED. The role of the c.m. energy dependence of the relativistic interquark
confining potential is emphasized and the structure of the Hamiltonian, to
order $1/c^2$, corresponding to confining scalar potentials, is displayed.
