After refreshing the well-known works by Mulase, Reyman and Semenov-Tian-Shanski on the algebraic integration of the Kadomtsev-Petviashvili (KP) hierarchy via r-matrices, we show in this talk:
– firstly how the classical algebra of  pseudo-differential operators over $S^1$ present in the formulation of the KP hierarchy, which coincide with the formal part of the so-called odd class (non formal) operators defined by Kontsevich and Vishik, can be extended to the class of formal classical pseudo-differential operators. For this, we describe the link between these two algebras which stands in an almost complex integrable structure. This part is based on a joint work with V. Rubtsov.
– secondly how the category of Frölicher spaces, which is a subcategory of diffeological spaces, can apply here to state rigorously smooth dependence of the solution of the KP hierarchy on the initial condition.  This part is mostly based on a joint work with Enrique G. Reyes on the « classical » KP hierarchy, which applies also to KP hierarchy in an « extended » class of operators of previously exposed.
https://indico.math.cnrs.fr/event/6567/