  In a previous paper we proved that after a simple transformation the
generating series of the linear Hodge integrals on the moduli space of stable
curves satisfies the hierarchy of the Intermediate Long Wave equation. In this
paper we present a much shorter proof of this fact. Our new proof is based on
an explicit formula for the one-point linear Hodge integrals that was found
independently by Faber, Pandharipande and Ekedahl, Lando, Shapiro, Vainshtein.
