  We reconsider the homogeneous Faddeev-Merkuriev integral equations for
three-body Coulombic systems with attractive Coulomb interactions and point out
that the resonant solutions are contaminated with spurious resonances. The
spurious solutions are related to the splitting of the attractive Coulomb
potential into short- and long-range parts, which is inherent in the approach,
but arbitrary to some extent. By varying the parameters of the splitting the
spurious solutions can easily be ruled out. We solve the integral equations by
using the Coulomb-Sturmian separable expansion approach. This solution method
provides an exact description of the threshold phenomena. We have found several
new S-wave resonances in the e- e+ e- system in the vicinity of thresholds.
