It is known that a generating function of the linear Hodge integrals is a tau function of the KP hierarchy, namely a oneparameter deformation of the KontsevichWitten taufunction (see Kazarian). Are there known any generating functions of the multiple Hodge integrals (integrals with multiple $\lambda$classes) with nice integrable properties?

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 (1partition and 2partition triple Hodge integrals) are found to be related to KP and 2Toda 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 3KP hierarchy. See also Mariño's book on ChernSimons, Matrix models and Topstrings about it. Slightly switching to autobiography, a somewhat different interpretation of the multiHodge integrals generating functions as dispersive deformations of the WittenKontsevich tau funcion is alluded to here. This is closer in spirit to the DubrovinZhang study of 1+1 integrable hierarchies of topological type. 

