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... the various intermediate continuum limits, as well as reductions to lower dimensional equations, in a systematic way. 2 Cauchy matrix scheme We will develop now a scheme along the lines of the paper =-=[19, 20]-=- where Atkinson and one of the present authors developed the elliptic soliton solutions for the ABS (Adler/Bobenko/Suris, cf. [4]) list of two-dimensional lattice equations, based on elliptic Cauchy m...