It is known that a generating function of the linear Hodge integrals is a tau function of the KP hierarchy, namely a one-parameter deformation of the Kontsevich-Witten tau-function  (see [Kazarian][1]). Are there known any generating functions of the multiple Hodge integrals (integrals with multiple $\lambda$-classes) with nice integrable properties?

  [1]: http://arxiv.org/abs/0809.3263