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The method due to Nijhoff and Bobenko & Suris to derive Lax pairs for partialdifference equations (PDeltaEs) is applied to edge constrained Boussinesqsystems. These systems are defined on a quadrilateral. They are consistentaround the cube but they contain equations defined on the edges of thequadrilateral.  By properly incorporating the edge equations into the algorithm, it isstraightforward to derive Lax matrices of minimal size. The 3 by 3 Lax matricesthus obtained are not unique but shown to be gauge-equivalent. The gaugematrices connecting the various Lax matrices are presented. It is also shownthat each of the Boussinesq systems admits a 4 by 4 Lax matrix. For eachsystem, the gauge-like transformations between Lax matrices of different sizesare explicitly given. To illustrate the analogy between continuous and latticesystems, the concept of gauge-equivalence of Lax pairs of nonlinear partialdifferential equations is briefly discussed.  The method to find Lax pairs of PDeltaEs is algorithmic and is beingimplemented in Mathematica. The Lax pair computations for this chapter helpedfurther improve and extend the capabilities of the software under development.

Lax Pairs for Edge-constrained Boussinesq Systems of Partial Difference Equations