  An approach based on combined solutions of the Bethe-Salpeter (BS) and
Dyson-Schwinger (DS) equations within the ladder-rainbow approximation in the
presence of singularities is proposed to describe the meson spectrum as quark
antiquark bound states. We consistently implement into the BS equation the
quark propagator functions from the DS equation, with and without pole-like
singularities, and show that, by knowing the precise positions of the poles and
their residues, one is able to develop reliable methods of obtaining finite
interaction BS kernels and to solve the BS equation numerically. We show that,
for bound states with masses $M < 1$ GeV, there are no singularities in the
propagator functions when employing the infrared part of the Maris-Tandy kernel
in truncated BS-DS equations. For $M >1 $ GeV, however, the propagator
functions reveal pole-like structures. Consequently, for each type of mesons
(unflavored, strange and charmed) we analyze the relevant intervals of $M$
where the pole-like singularities of the corresponding quark propagator
influence the solution of the BS equation and develop a framework within which
they can be consistently accounted for. The BS equation is solved for
pseudo-scalar and vector mesons. Results are in a good agreement with
experimental data. Our analysis is directly related to the future physics
programme at FAIR with respect to open charm degrees of freedom.
