  We review the integration of the KP hierarchy in several non-standard
contexts. Specifically, we consider KP in the following associative
differential algebras: an algebra equipped with a nilpotent derivation; an
algebra of functions equipped with a derivation that generalizes the gradient
operator; an algebra of quaternion-valued functions; a differential Lie
algebra; an algebra of polynomials equipped with the Pincherle differential; a
Moyal algebra. In all these cases we can formulate and solve the Cauchy problem
of the KP hierarchy. Also, in each of these cases we derive different
Zakharov-Shabat $(t_2,t_3)$-equations -- that is, different
Kadomtsev-Petviashvili equations -- and we prove existence of solutions arising
from solutions to the corresponding KP hierarchy.
