  The nonlinear structures in 2D quantum gravity coupled to the $(q+1,q)$
minimal model are studied in the Liouville theory to clarify the factorization
and the physical states. It is confirmed that the dressed primary states
outside the minimal table are identified with the gravitational descendants.
Using the discrete states of ghost number zero and one we construct the
currents and investigate the Ward identities which are identified with the W
and the Virasoro constraints. As nontrivial examples we derive the $L_0$, $L_1$
and $W_{-1}^{(3)}$ equations exactly. $L_n$ and $W^{(k)}_n$ equations are also
discussed. We then explicitly show the decoupling of the edge states $O_j
~(j=0~ {\rm mod}~ q) $. We consider the interaction theory perturbed by the
cosmological constant $O_1$ and the screening charge $S^+ =O_{2q+1}$. The
formalism can be easily generalized to potential models other than the
screening charge.
