2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/101033The Extended Toda Hierarchy (shortly ETH) was introduce by E. Getzler \cite{Ge} and independently by Y. Zhang \cite{Z} in order to describe an integrable hierarchy which governs the Gromov--Witten invariants of $\C P^1$. The {\em Lax type} presentation of the ETH was given in \cite{CDZ}. In this paper we give a description of the ETH in terms of {\em tau-functions} and Hirota Quadratic Equations (known also as Hirota Bilinear Equations). A new feature here is that the Hirota equations are given in terms of vertex operators taking values in the algebra of differential operators on the affine line.15 pages, pending final approval in Duke, this replacement is the first part of the old versionAlgebraic Geometrytext