  In this work we compute bound solutions for particles and anti-particles of
the Dirac equation for a pure tensor radial Coulomb potential plus a constant.
We find that the binding depends on the sign the tensor constant potential, and
allows only bound solutions for a certain sign and magnitude of the $\kappa$
quantum number, which is related to the spin-orbit coupling in the Dirac
equation. This relation is reversed for anti-particle solutions. On the other
hand, the Coulomb tensor potential, although not biding, changes the range of
$\kappa$ values for which there are bound solutions.
