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). Are there known any generating functions of the multiple Hodge integrals (integrals with multiple $\lambda$-classes) with nice integrable properties?
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There's been a lot of interest on triple Hodge integrals in the last 10 years, mostly after the work of Mariño and Vafa on framed knots (see here). The question of integrability was first studied, as far as I am aware, by Jian Zhou in his paper on Hodge Integrals and Integrable Hierarchies, where two subcases (1-partition and 2-partition triple Hodge integrals) are found to be related to KP and 2-Toda respectively. The general case is considered by Aganagic, Dijkgraaf, Klemm, Mariño and Vafa in their Topological Strings and Integrable Hierarchies paper; the relevant integrable hierarchy is the 3-KP hierarchy. See also Mariño's book on Chern-Simons, Matrix models and Topstrings about it.
Slightly switching to autobiography, a somewhat different interpretation of the multi-Hodge integrals generating functions as dispersive deformations of the Witten-Kontsevich tau funcion is alluded to here. This is closer in spirit to the Dubrovin-Zhang study of 1+1 integrable hierarchies of topological type.
