Direct linearization approach to discrete integrable systems associated with Z_N graded Lax pairs

Fordy and Xenitidis [J. Phys. A: Math. Theor. 50 (2017) 165205. (doi:10.1088/1751-8121/aa639a)] recently proposed a large class of discrete integrable systems which include a number of novel integrable difference equations, from the perspective of Z_N graded Lax pairs, without providing solutions. In this paper, we establish the link between the Fordy–Xenitidis (FX) discrete systems in coprime case and linear integral equations in certain form, which reveals solution structure of these equations. The bilinear form of the FX integrable difference equations is also presented together with the associated general tau function. Furthermore, the solution structure explains the connections between the FX novel models and the discrete Gel’fand–Dikii hierarchy.