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