  Following Natanzon-Zabrodin, we explore the Kadomtsev-Petviashvili hierarchy
as an infinite system of mutually consistent relations on the second
derivatives of the free energy with some universal coefficients. From this
point of view, various combinatorial properties of these coefficients naturally
highlight certain non-trivial properties of the KP hierarchy. Furthermore, this
approach allows us to suggest several interesting directions of the KP
deformation via a deformation of these coefficients. We also construct an
eigenvalue matrix model, whose correlators fully describe the universal KP
coefficients, which allows us to further study their properties and
generalizations.
