  The Lax pair formulation of the two dimensional induced gravity in the
light-cone gauge is extended to the more general $w_N$ theories. After
presenting the $w_2$ and $w_3$ gravities, we give a general prescription for an
arbitrary $w_N$ case. This is further illustrated with the $w_4$ 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 presence of the constraints are also pointed out and
are contrasted with those of the ordinary 2d-gravity.
