On a direct procedure for the disclosure of Lax pairs and Bäcklund transformations

A direct and unifying scheme for the disclosure of bilinear Bäcklund transformations and linear Lax systems associated with soliton equations is presented. The scheme is based on a concept of scale invariance and on the use of a class of partitional polynomials: the binary Bell polynomials. The applicability of the procedure is tested on a variety of soliton equations.