  We present the Lax pair formalism for certain extension of the continuous
limit of the classical Toda lattice hierarchy, provide a well defined notion of
tau function for its solutions, and give an explicit formulation of the
relationship between the $CP^1$ topological sigma model and the extended Toda
hierarchy. We also establish an equivalence of the latter with certain
extension of the nonlinear Schr\"odinger hierarchy.

Exactly Solvable and Integrable Systems;High Energy Physics - Theory;Differential Geometry