  In view of the mass spectrum of heavy mesons in vacuum the analytical
properties of the solutions of the truncated Dyson-Schwinger equatio for the
quark propagator within the rainbow approximation are analysed in some detail.
In Euclidean space, the quark propagator is not an analytical function
possessing, in general, an infinite number of singularities (poles) which
hamper to solve the Bethe-Salpeter equation. However, for light mesons (with
masses M_{q\bar q} <= 1 GeV) all singularities are located outside the region
within which the Bethe-Salpeter equation is defined. With an increase of the
considered meson masses this region enlarges and already at masses >= 1 GeV,
the poles of propagators of u,d and s quarks fall within the integration domain
of the Bethe-Salpeter equation. Nevertheless, it is established that for meson
masses up to M_{q\bar q}~=3 GeV only the first, mutually complex conjugated,
poles contribute to the solution. We argue that, by knowing the position of the
poles and their residues, a reliable parametrisation of the quark propagators
can be found and used in numerical procedures of solving the Bethe-Salpeter
equation. Our analysis is directly related to the future physics programme at
FAIR with respect to open charm degrees of freedom.
