  Let A be a nonassociative algebra such that the associator (A,A^2,A)
vanishes. If A is freely generated by an element f, there are commuting
derivations delta_n, n=1,2,..., such that delta_n(f) is a nonlinear homogeneous
polynomial in f of degree n+1. We prove that the expressions delta_{n_1} ...
delta_{n_k}(f) satisfy identities which are in correspondence with the
equations of the Kadomtsev-Petviashvili (KP) hierarchy. As a consequence,
solutions of the `nonassociative hierarchy' partial_{t_n}(f) = delta_n(f),
n=1,2,..., of ordinary differential equations lead to solutions of the KP
hierarchy. The framework is extended by introducing the notion of an A-module
and constructing, with the help of the derivations delta_n, zero curvature
connections and linear systems.
