  Given a pair $(\Gamma,\rho)$ of a Fuchsian group of the first kind, and a
unitary representation $\rho$ of $\Gamma$ of arbitrary rank, the problem of
construction of vector-valued Poincar\'e series of weight 2 is considered.
Implications in the theory of parabolic bundles are discussed. When the genus
of the group is zero, it is shown how an explicit basis for the space of these
functions can be constructed.
