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String equations of the $p$-th generalized Kontsevich model and thecompactified $c = 1$ string theory are re-examined in the language of the Todalattice hierarchy. As opposed to a hypothesis postulated in the literature, thegeneralized 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 Todalattice hierarchy including these models are constructed, and shown to satisfygeneralized string equations. The status of a variety of $c \le 1$ stringmodels is discussed in this new framework.Comment: 35pages, LaTeX Errors are corrected in Eqs. (2.21), (2.36), (2.33),  (3.3), (5.10), (6.1), sentences after (3.19) and theorem 5. A few references  are update

Toda lattice hierarchy and generalized string equations