  In this paper we introduce new various generalizations of the classical
Kadomtsev-Petviashvili hierarchy in the case of operators in several variables.
These generalizations are the candidates for systems that should play the role,
analogous to the role of the KP hierarchy in the classical KP theory, in a
generalized KP theory. In particular, they should describe flows of some
generalized geometric datas, including those described by A.N.Parshin, for
certain initial conditions. The unique solvability of the initial value problem
for the generalized KP hierarchies is established. The connection of these
systems with universal families of isospectral deformations of certain pairs of
commuting differential operators is opened. To prove the solvability of the
systems we generalize several results from the works of M.Mulase and
A.N.Parshin.
