  We construct an integrable hierarchy in terms of vertex operators and Hirota
Quadratic Equations (HQE shortly) and we show that the equivariant total
descendant potential of $\C P^1$ satisfies the HQE. Our prove is based on the
quantization formalism developed in \cite{G1}, \cite{G2}, and on the
equivariant mirror model of $\C P^1.$ The vertex operators in our construction
obey certain transformation law under change of coordinates, which might be
important for generalizing the HQE to other manifolds.
  We also show that under certain change of the variables, which is due to E.
Getzler, the HQE are transformed into the HQE of the 2-Toda hierarchy. Thus we
obtain a new proof of the equivariant Toda conjecture.
