A systematic, elementary and pedagogical approach to a class of soliton equations, and to their spectral formulation, is presented. This approach, based on the use of exponential polynomials, follows naturally from a comparison of some simple results for two representatives of the class: the KdV-and the Boussinesq-equation.

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Classical Darboux transformations together with partitional polynomials are used as elementary tools for the construction of members of a basic hierarchy of integrable nonlinear partial differential equations (the KP hierarchy).

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A generic formula is presented which relates th e H irota D -operators to simple com binatorics. P articular classes of p artitio n polynom ials (Bell-polynomials and general izations) are found to play an im p o rtan t role in the characterization of bilinearizable equations. As a consequence it is shown th a t bilinear Backlund transform ations for single-field bilinearizable equations linearize system atically into corresponding Laxpairs. p=0'-P '