When considering the classification problem of integrable systems, we found a class of deformations of the KdV hierarchy which contains infinitely many parameters. It is conjectured that this integrable hierarchy is just the one that governs the generating function of all Hodge integrals. The generating function of special cubic Hodge integrals is much more important than the other cases, since it is related to the localization computation of Gromov-Witten invariants for a toric Calabi-Yau threefold. We formulate a conjecture on the integrable hierarchy that governs it. Speaker(s): Si-Qi Liu (Tsinghua University)