  We calculate resonances in three-body systems with attractive Coulomb
potentials by solving the homogeneous Faddeev-Merkuriev integral equations for
complex energies. The equations are solved by using the Coulomb-Sturmian
separable expansion approach. This approach provides an exact treatment of the
threshold behavior of the three-body Coulombic systems. We considered the
negative positronium ion and, besides locating all the previously know $S$-wave
resonances, we found a whole bunch of new resonances accumulated just slightly
above the two-body thresholds. The way they accumulate indicates that probably
there are infinitely many resonances just above the two-body thresholds, and
this might be a general property of three-body systems with attractive Coulomb
potentials.
