Lax-pair formulation of the w-gravity theories in two dimensions

The Lax-pair formulation of the two-dimensional induced gravity in the light-cone gauge is extended to the more general wN theories. After presenting the w2 and w3 gravities, we give a general prescription for an arbitrary wN case. This is further illustrated with the w4 gravity to point out some peculiarities. The constraints and the possible presence of the cosmological constants are systematically exhibited in the zero-curvature condition, which also yields the relevant Ward identities. The restrictions on the gauge parameters in the presence of the constraints are pointed out too, and are contrasted with those of the ordinary 2D gravity.