  A first attempt at adding matter degrees of freedom to the two-dimensional
``vacuum'' gravity model presented in gr-qc/9907071 is analyzed in this paper.
Just as in the previous pure gravity case, quantum diffeomorphism operators
(constructed from a Virasoro algebra) possess a dynamical content; their gauge
nature is recovered only after the classical limit. Emphasis is placed on the
new physical modes modelled on a SU(1,1)-Kac-Moody algebra. The non-trivial
coupling to ``gravity'' is a consequence of the natural semi-direct structure
of the entire extended algebra. A representation associated with the discrete
series of the rigid SU(1,1) is revisited in the light of previously neglected
crucial global features which imply the appearance of an SU(1,1)-Kac-Moody
fusion rule, determining the rather entangled quantum structure of the physical
system. In the classical limit, an action which explicitly couples gravity and
matter modes governs the dynamics.
