The Prolongation Structures of Quasi-Polynomial Flows

We look closely at the process of finding a Wahlquist-Estabrook prolongation structure for a given (system of) nonlinear evolution equation(s). There are two main steps in this calculation: the first, to reduce the problem to the investigation of a finitely generated, free Lie algebra with constraints; the second, to find a finite-dimensional linear representation of these generators. We discuss some of the difficulties that arise in this calculation. For quasi-polynomial flows (defined later) we give an algorithm for the first step. We do not totally solve the problems of the second step, but do give an algebraic framework and a number of techniques that are quite generally applicable. We illustrate these methods with many examples, several of which are new.