  We present some explicit results on the structure of singular vectors in
$c=2$ Verma modules of the $\cW_3$ algebra. Using the embedding patterns of
those vectors we construct resolutions for the $c=2$ irreducible modules, and
thus are able to compute some of the BRST cohomology of $\cW_3$ gravity coupled
to $c=2$ matter. In particular, we determine the states in the ground ring of
the theory. (To appear in the proceedings of the AMS Special Session on
"Geometry and Physics", USC, Los Angeles, November 5-6, 1992)
