  String equations of the $p$-th generalized Kontsevich model and the
compactified $c = 1$ string theory are re-examined in the language of the Toda
lattice hierarchy. As opposed to a hypothesis postulated in the literature, the
generalized Kontsevich model at $p = -1$ does not coincide with the $c = 1$
string theory at self-dual radius. A broader family of solutions of the Toda
lattice hierarchy including these models are constructed, and shown to satisfy
generalized string equations. The status of a variety of $c \le 1$ string
models is discussed in this new framework.
