  The method due to Nijhoff and Bobenko & Suris to derive Lax pairs for partial
difference equations (PDeltaEs) is applied to edge constrained Boussinesq
systems. These systems are defined on a quadrilateral. They are consistent
around the cube but they contain equations defined on the edges of the
quadrilateral.
  By properly incorporating the edge equations into the algorithm, it is
straightforward to derive Lax matrices of minimal size. The 3 by 3 Lax matrices
thus obtained are not unique but shown to be gauge-equivalent. The gauge
matrices connecting the various Lax matrices are presented. It is also shown
that each of the Boussinesq systems admits a 4 by 4 Lax matrix. For each
system, the gauge-like transformations between Lax matrices of different sizes
are explicitly given. To illustrate the analogy between continuous and lattice
systems, the concept of gauge-equivalence of Lax pairs of nonlinear partial
differential equations is briefly discussed.
  The method to find Lax pairs of PDeltaEs is algorithmic and is being
implemented in Mathematica. The Lax pair computations for this chapter helped
further improve and extend the capabilities of the software under development.
