nd discusses the general formulation of these transformations for any dimension $N$.

$\boldsymbol{{\mathbb{Z}}_N}$-graded Lax pairs {#sec:ZN-LP}
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We now consider the specific discrete Lax pairs, which we introduced in [@f14-3; @f17-2]. Consider a pair of matrix equations of the form

\[eq:dLP-gen\] $$\begin{gathered}
 \Psi_{m+1,n} = L_{m,n} \Psi_{m,n} \equiv \big( U_{m,n} + \lambda \Omega^{\ell_1}\big) \Psi_{m,n}, \label{eq:dLP-gen-L} \\
 \Psi_{m,n+1} = M_{m,n} \Psi_{m,n} \equiv \big( V_{m,n} + \lambda \Omega^{\ell_2}\big) \Psi_{m,n}, \label{eq:dLP-gen-M}\end{gathered}$$ where $$\begin{gathered}
 \label{eq:A-B-entries}
U_{m,n} = \operatorname{diag}\big(u^{(0)}_{m,n},\dots,u^{(N-1)}_{m,n}\big) \Omega^{k_1},\qquad
 V_{m,n} = \operatorname{diag}\big(v^{(0)}_{m,n},\dots,v^{(N-1)}_{m,n}\big) \Omega^{k_2},\end{gathered}$$ and $$\begin{gathered}
(\Omega)_{i,j} = \delta_{j-i,1} + \delta_{i-j,N-1}.\end{gathered}$$

The matrix $\Omega$ defines a grading and the four matrices of (\[eq:dLP-gen\]) are said to be of respective levels $k_i$, $\ell_i$, with $\ell_i\neq k_i$ (for each $i$). The Lax pair is characterised by the quadruple $(k_1,\ell_1;k_2,\ell_2)$, which we refer to as [*the level structure*]{} of the system, and for consistency, we require $$\begin{gathered}
k_1 + \ell_2 \equiv k_2 + \ell_1 \quad (\bmod N).\end{gathered}$$ Since matrices $U$, $V$ and $\Omega$ are independent of $\lambda$, the compatibility condition of (\[eq:dLP-gen\]), $$\begin{gathered}
 L_{m,n+1} M_{m,n} = M_{m+1,n} L_{m,n},\end{gathered}$$ splits into the system

\[eq:dLP-gen-scc\] $$\begin{gathered}
U_{m,n+1} V_{m,n} = V_{m+1,n} U_{m,n} , \label{eq:dLP-gen-scc-1}\\
U_{m,n+1} \Omega^{\ell_2} - \Omega^{\ell_2} U_{m,n} = V_{m+1,n} \Omega^{\ell_1} - \Omega^{\ell_1} V_{m,n}, \label{eq:dLP-gen-scc-2}\end{gathered}$$

which can be written explicitly as

\[eq:dLP-ex-cc\] $$\begin{gathered}
u^{(i)}_{m,n+1} v_{m,n}^{(i+k_1)} = v^{(i)}_{m+1,n} u^{(i+k_2)}_{m,n} , \label{eq:dLP-ex-cc-1}\\
u^{(i)}_{m,n+1