\section{Introduction}\label{Introduction}
In this paper, we focus on the following two integrable partial difference equations (P$\Delta$E s).
One is Hirota's discerete Korteweg-de Vries (dKdV) equation \cite{CNP1991:MR1111648,TGR2001:zbMATH01560028,kajiwara2008bilinearization}:
\begin{equation}\label{eqn:dKdV
 u_{l+1,m+1}-u_{l,m}
 =\dfrac{\beta_{m+1}-\alpha_l}{u_{l,m+1}}-\dfrac{\beta_m-\alpha_{l+1}}{u_{l+1,m}},
\end{equation}
where $(l,m)\in\mathbb{Z}^2$ is a lattice parameter, $u_{l,m}\in\mathbb{C}$ is its function and $\alpha_l,\beta_m\in\mathbb{C}$ are parameters that depend only on $l$ and $m$, respectively,
and the other is the lattice sine-Gordon (lsG) equation\cite{KN2018:zbMATH07031158}:
\begin{equation}\label{eqn:lsG
 \dfrac{x_{l+1,m+1}}{x_{l,m}}
 =\left(\dfrac{p_{l+1}-q_mx_{l+1,m}}{q_m-p_{l+1}x_{l+1,m}}\right)
 \left(\dfrac{q_{m+1}-p_lx_{l,m+1}}{p_l-q_{m+1}x_{l,m+1}}\right),
\end{equation}
where $(l,m)\in\mathbb{Z}^2$ is a lattice parameter, $x_{l,m}\in\mathbb{C}$ is its function and $p_l,q_m\in\mathbb{C}$ are parameters that depend only on $l$ and $m$, respectively.

The {P$\Delta$E} \eqref{eqn:dKdV} is an integrable non-autonomous discrete version of the Korteweg-de Vries (KdV) equation\cite{KDV1895:zbMATH02679684}:
\begin{equation}\label{eqn:KdV}
 w_t+6ww_x+w_{xxx}=0,
\end{equation}
where $(t,x)\in\mathbb{C}^2$ and $w=w(t,x)\in\mathbb{C}$,
which is known as a mathematical model of waves on shallow water surfaces.
On the other hand, the {P$\Delta$E} \eqref{eqn:lsG} is an integrable non-autonomous discrete version of the sine-Gordon equation:
\begin{equation}\label{eqn:sg}
 \phi_{tt}-\phi_{xx}+\sin{\phi}=0,
\end{equation}
where $(t,x)\in\mathbb{C}^2$ and $\phi=\phi(t,x)\in\mathbb{C}$,
which is known as a motion equation of a row of pendulums hanging from a rod and being coupled by torsion springs.

The consistency around a cube (CAC) property\cite{NQC1983:MR719638,NCWQ1984:MR763123,QNCL1984:MR761644,NS1998:zbMATH01844203,ABS2003:MR1962121,ABS2009:MR2503862,BollR2011:MR2846098,BollR2012:MR3010833,NW2001:MR1869690} is known as an integrability of quad-equations.
(For the CAC property, see Appendix \ref{subsection:CAC_def}.)
As classifications of quad-equations by using the CAC property, the lists of equations by Adler-Bobenko-Suris \cite{ABS2003:MR1962121,ABS2009:MR2503862}, Boll\cite{BollR2011:MR2846098,BollR2012:MR3010833} and Hietarinta \cite{HietarintaJ2019:zbMATH07053246} are known.
However, the P$\Delta$E s \eqref{eqn:dKdV} and \eqref{eqn:lsG} are not included in these lists, and the structures of their consistency have not been reported.

Recently, it has been shown that the P$\Delta$E s \eqref{eqn:dKdV} and \eqref{eqn:lsG} have the consistency around a broken cube (CABC) property\cite{JN2021:zbMATH07476241,nakazono2022properties}.
(For the CABC property, see Appendix \ref{subsection:CABC_def}.)
By considering the B\"acklund transformation (BT)
\begin{subequations}\label{eqns:dKdV_CABC_1
\begin{align}
 &u_{l,m}-v_{l+1,m+1}-\dfrac{\beta_m-\alpha_{l+1}}{u_{l+1,m}}+\dfrac{\gamma-\alpha_l}{v_{l,m+1}}=0,
 \label{eqn:dKdV_CABC_1_S}\\
 &(u_{l,m}-v_{l,m})\left(\dfrac{\beta_m-\alpha_l}{u_{l,m}}+v_{l,m+1}\right)-\beta_m+\gamma=0,
 \label{eqn:dKdV_CABC_1_B}\\
 &\dfrac{\beta_m-\alpha_l}{u_{l,m}}-\dfrac{\gamma-\alpha_l}{v_{l,m}}-u_{l+1,m}+v_{l+1,m}=0,
 \label{eqn:dKdV_CABC_1_C}
\end{align}
\end{subequations}
 from the {P$\Delta$E} \eqref{eqn:dKdV} to the multi-quadratic equation
\begin{equation}\label{eqn:multiquadratic
\begin{split}
 &\beta_m\left(v_{l,m}\,v_{l+1,m}-v_{l,m+1}\,v_{l+1,m+1}\right)^2\\
 &\quad +\bigl(v_{l,m}-v_{l+1,m+1}\bigr)\bigl(v_{l+1,m}-v_{l,m+1}\bigr)\bigl(\gamma-v_{l,m}\,v_{l+1,m}\bigr)\bigl(\gamma-v_{l,m+1}\,v_{l+1,m+1}\bigr)\\
 &\quad +\bigl(\alpha_{l+1}v_{l,m}-\alpha_lv_{l+1,m+1}\bigr)\bigl(\alpha_lv_{l+1,m}-\alpha_{l+1}v_{l,m+1}\bigr)\\
 &\quad -(\alpha_l+\alpha_{l+1})\bigl(\gamma\,v_{l,m}\,v_{l+1,m}+\gamma\,v_{l,m+1}\,v_{l+1,m+1}-2\,v_{l,m}\,v_{l+1,m}\,v_{l,m+1}\,v_{l+1,m+1}\bigr)\\
 &\quad+\bigl(\alpha_{l+1}\,v_{l,m}\,v_{l,m+1}+\alpha_l\,v_{l+1,m}\,v_{l+1,m+1}\bigr)\bigl(2\gamma - v_{l,m} v_{l+1,m}-v_{l,m+1} \,v_{l+1,m+1}\bigr) =0,
\end{split}
\end{equation}
where $\gamma\in\mathbb{C}$ is a constant parameter, the CABC property of the {P$\Delta$E} \eqref{eqn:dKdV} was shown in \cite{JN2021:zbMATH07476241}.
Similarly, by considering the following BT from the {P$\Delta$E} \eqref{eqn:lsG} to Equation \eqref{eqn:multiquadratic}:
\begin{subequations}\label{eqns:lsG_CABC
\begin{align}
 &\dfrac{v_{l,m+1}v_{l+1,m+1}}{p_lp_{l+1}}-\dfrac{x_{l,m}(p_{l+1}-q_m x_{l+1,m})}{p_{l+1}x_{l+1,m}-q_m}=0,
 \label{eqn:lsG_CABC_S}\\
 &\dfrac{v_{l,m}v_{l,m+1}}{{p_l}^2}-\dfrac{x_{l,m}(p_l-q_mx_{l,m})}{p_lx_{l,m}-q_m}=0,
 \label{eqn:lsG_CABC_B}\\
 &\dfrac{v_{l,m}v_{l+1,m}}{p_lp_{l+1}}-\dfrac{x_{l+1,m}(p_l-q_mx_{l,m})}{p_lx_{l,m}-q_m}=0,
 \label{eqn:lsG_CABC_C}
\end{align}
\end{subequations}
the CABC property of the {P$\Delta$E} \eqref{eqn:lsG} was shown in \cite{nakazono2022properties}.
In this case, the relation between the parameters $p_l$, $q_m$, $\alpha_l$, $\beta_m$ and $\gamma$ is given by
\begin{equation}\label{eqn:pq_albe_ga
 p_l=\sqrt{\alpha_l-\gamma},\quad
 q_m=\sqrt{\beta_m-\gamma}.
\end{equation}
Since having a CABC property does not imply not having a CAC property, the CABC property of the P$\Delta$E s \eqref{eqn:dKdV} and \eqref{eqn:lsG} was shown, but whether they have the CAC property or not is an open problem.

\begin{remark}
The first condition in Equation \eqref{eqn:pq_albe_ga} means that given the complex parameters $\alpha_l$ and $\gamma$, we determine one value of the complex parameter $p_l$ that satisfies 
\begin{equation}
 {p_l}^2=\alpha_l-\gamma.
\end{equation}
The same is true for the second condition in Equation \eqref{eqn:pq_albe_ga}.
In addition, if the root symbol $\sqrt{~\,~}$ is used in the following, it means the same thing.
\end{remark}

In this paper, we introduce another CABC-type BT from the {P$\Delta$E} \eqref{eqn:dKdV} to Equation \eqref{eqn:multiquadratic}, which is different from the CABC-type BT \eqref{eqns:dKdV_CABC_1}. 
Then, using a total of three CABC-type BTs, we show the following results:
\begin{enumerate}
\item[(i)] 
derive two CAC-type BTs from the {P$\Delta$E} \eqref{eqn:dKdV} to the {P$\Delta$E} \eqref{eqn:lsG};
\item[(ii)]  
derive a CAC-type auto-B\"acklund transformation (auto-BT) of the {P$\Delta$E} \eqref{eqn:lsG};
\item[(iii)] 
construct Lax pairs of the P$\Delta$E s \eqref{eqn:dKdV} and \eqref{eqn:lsG};
\item[(iv)] 
derive integrable systems of P$\Delta$E s on the 3-dimensional integer lattice containing the P$\Delta$E s \eqref{eqn:dKdV} and \eqref{eqn:lsG}.
\end{enumerate}

\subsection{Notation and Terminology}\label{subsection:NT}
\begin{itemize}
\item 
For simplicity, we use the following shorthand notations:
\begin{equation}
 \overline{\rule{0em}{0.5em}\hspace{0.6em}}:\,l\to l+1,\quad
 \widetilde{}\,:\,m\to m+1,\quad
 \widehat{}\,:\,n\to n+1.
\end{equation}
For example, 
\begin{equation}
 \overline{u}=u_{l+1,m},\quad
 \widetilde{u}=u_{l,m+1},\quad
 \widetilde{\ou}=u_{l+1,m+1},\quad
 \overline{\overline{u}}=u_{l+2,m},\quad
 \widetilde{\widetilde{u}}=u_{l,m+2}
\end{equation}
for $u=u_{l,m}$ and 
\begin{equation}
 \overline{u}=u_{l+1,m,n},\quad
 \widetilde{u}=u_{l,m+1,n},\quad
 \widehat{u}=u_{l,m,n+1},\quad
 \widehat{\wou}=u_{l+1,m+1,n+1}
\end{equation}
for $u=u_{l,m,n}$, and the like.

\item
If there is a substitution notation (e.g., $l\to l+1$) in the subscript of an equation number, it means the equation with the substitution applied to all symbols in the corresponding equation.
For example, \eqref{eqn:dKdV}$_{l\to l+1}$ implies the following equation:
\
 u_{l+2,m+1}-u_{l+1,m}
 =\dfrac{\beta_{m+1}-\alpha_{l+1}}{u_{l+1,m+1}}-\dfrac{\beta_m-\alpha_{l+2}}{u_{l+2,m}}.
\]

\item
The following terminologies are used in this paper.
\begin{enumerate}
\item 
A multivariate polynomial with complex coefficients that is affine linear with respect to each variable is called a {\it multilinear polynomial}.
For example, the general form of multilinear polynomial in four variables is given by
\begin{align}
 &A_1xyzw+A_2xyz+A_3xyw+A_4xzw+A_5yzw+A_6xy+A_7xz+A_8xw\notag\\
 &\quad +A_9yz+A_{10}yw+A_{11}zw+A_{12}x+A_{13}y+A_{14}z+A_{15}w+A_{16},
\end{align}
where $A_i\in\mathbb{C}$ are parameters.
\item 
Let $Q(x,y,z,w)$ be a multilinear polynomial and $x=f(y,z,w)$ be the solution of $Q(x,y,z,w)=0$.
We call $Q(x,y,z,w)$ an {\it irreducible multilinear polynomial} in four variables if the following holds:
\begin{equation}
 \dfrac{\partial}{\partial y} f(y,z,w)\neq0,\quad
 \dfrac{\partial}{\partial z} f(y,z,w)\neq0,\quad
 \dfrac{\partial}{\partial w} f(y,z,w)\neq0.
\end{equation}
\item
Consider a {P$\Delta$E} of the form 
\begin{equation}\label{eqn:general_quad_eqn}
 Q(u,\overline{u},\widetilde{u},\widetilde{\ou}\,)=0,
\end{equation}
where $u=u_{l,m}$ and $Q$ is an irreducible multilinear polynomial. 
By putting the variable $u_{l,m}$ on the grid point $(l,m)$ of the integer lattice $\mathbb{Z}^2$,
the {P$\Delta$E} \eqref{eqn:general_quad_eqn} can be regarded as a relation on the quadrilateral given by
\[
 (l,m),\quad (l+1,m),\quad (l,m+1),\quad (l+1,m+1)
\]
as shown in Figure \ref{fig:quadrilateral}.
For this reason, a {P$\Delta$E} of the form \eqref{eqn:general_quad_eqn} is called a {\it quad-equation}.
Note that quad-equations are not necessarily autonomous, but they may contain parameters that evolve with $l$ or $m$.
\end{enumerate}
\end{itemize}

\begin{figure}[htbp]
\begin{center}
 \includegraphics[width=0.35\textwidth]{fig_quadrilateral.eps}
\end{center}
\caption{A quadrilateral for the quad-equation \eqref{eqn:general_quad_eqn}.}
\label{fig:quadrilateral}
\end{figure}

\subsection{Plan of the paper}
This paper is organized as follows.
In \S \ref{section:dKdV_lsG_consistency}, using the CABC property of the dKdV equation \eqref{eqn:dKdV} and the lsG equation \eqref{eqn:lsG}, we show that they have the CAC property.
In \S \ref{section:3D}, we show how the P$\Delta$E s \eqref{eqn:dKdV} and \eqref{eqn:lsG} can be extended to systems of P$\Delta$E s on the 3-dimensional integer lattice.
Some concluding remarks are given in \S \ref{ConcludingRemarks}.
In Appendix \ref{section:CAC_CABC}, we recall the definitions of the CAC and CABC properties.
In Appendix \ref{section:Lax}, we list the Lax pairs of the P$\Delta$E s \eqref{eqn:dKdV} and \eqref{eqn:lsG} constructed in this study.
In Appendix \ref{section:proof_tau}, we give the proof of Theorem \ref{theorem:tau}.
\section{Consistencies of the dKdV equation and the lsG equation}\label{section:dKdV_lsG_consistency}
In this section, we give the new consistencies of the dKdV equation \eqref{eqn:dKdV} and the lsG equation \eqref{eqn:lsG}.
See Appendix \ref{section:CAC_CABC} for descriptions of the CABC and CAC properties we deal with here.

\subsection{A CABC-type BT from the {P$\Delta$E} \eqref{eqn:dKdV} to the multi-quadratic equation \eqref{eqn:multiquadratic}}\label{subsection:dKdV_CABC}
The following system of equations is a new CABC-type BT from the {P$\Delta$E} \eqref{eqn:dKdV} to Equation \eqref{eqn:multiquadratic}:
\begin{subequations}\label{eqns:dKdV_CABC_2
\begin{align}
 &u-\widetilde{v}-\dfrac{\beta_m-\alpha_{l+1}}{\overline{u}}+\dfrac{\gamma-\alpha_{l+1}}{\widetilde{\ov}}=0,
 \label{eqn:dKdV_CABC_2_S}\\
 &(u-\widetilde{v})\left(\dfrac{\beta_m-\alpha_l}{u}+v\right)-\beta_m+\gamma=0,
 \label{eqn:dKdV_CABC_2_B}\\
 &\dfrac{\beta_m-\alpha_l}{u}-\dfrac{\gamma-\alpha_{l+1}}{\overline{v}}-\overline{u}+v=0,
 \label{eqn:dKdV_CABC_2_C}
\end{align}
\end{subequations}
where $u=u_{l,m}$ and $v=v_{l,m}$.
By direct calculation, we can verify that the system of equations \eqref{eqn:dKdV} and \eqref{eqns:dKdV_CABC_2} has the CABC and tetrahedron properties.
Indeed, the following relations can be derived from the CABC-system \eqref{eqn:dKdV} and \eqref{eqns:dKdV_CABC_2}:
\begin{equation
 \dfrac{\alpha_{l+1}-\gamma}{\widetilde{\ov}}=\dfrac{\alpha_{l+1}-\beta_m}{\overline{u}}-\dfrac{(\beta_m-\gamma)u}{\alpha_l-\beta_m-uv},\quad
 \widetilde{v}=u+\dfrac{(\beta_m-\gamma)\overline{v}}{\alpha_{l+1}-\gamma-\overline{u}\,\overline{v}}.
\end{equation}
As shown in \cite{JN2021:zbMATH07476241,nakazono2022properties}, the Lax pair \eqref{eqn:Lax_phi} with \eqref{eqn:Lax_dKdV_CABC} of the {P$\Delta$E} \eqref{eqn:dKdV} can be constructed from the equations \eqref{eqn:dKdV_CABC_2_B} and \eqref{eqn:dKdV_CABC_2_C} by setting
\begin{equation}
 v_{l,m}=\dfrac{F_{l,m}}{G_{l,m}},\quad
 \Phi_{l,m}=\begin{pmatrix}F_{l,m}\\G_{l,m}\end{pmatrix}.
\end{equation}
\subsection{A CAC-type BT from the {P$\Delta$E} \eqref{eqn:dKdV} to the {P$\Delta$E} \eqref{eqn:lsG} (I)}\label{subsection:CABC_to_CAC_1}
In this subsection, using the CABC-type BTs \eqref{eqns:dKdV_CABC_1} and \eqref{eqns:lsG_CABC}, we obtain a CAC-type BT from the {P$\Delta$E} \eqref{eqn:dKdV} to the {P$\Delta$E} \eqref{eqn:lsG}.

Eliminating $\widetilde{v}$ from the equations \eqref{eqn:dKdV_CABC_1_B} and \eqref{eqn:lsG_CABC_B}, we obtain 
\begin{equation}\label{eqn:v_ux_CABC_1
 \Big((q_m-p_lx)v+p_lux\Big)
 \Big(({p_l}^2-{q_m}^2)v-p_lu(p_l-q_mx)\Big)
 =0.
\end{equation}
Let us assume that the following holds:
\begin{equation}\label{eqn:v_ux_CABC_2
 v=\dfrac{p_lux}{p_lx-q_m},
\end{equation}
which satisfies Equation \eqref{eqn:v_ux_CABC_1}.
Eliminating $v$ from Equation \eqref{eqn:dKdV_CABC_1_B} by using Equation \eqref{eqn:v_ux_CABC_2}
and $\widetilde{v}$ from Equation \eqref{eqn:dKdV_CABC_1_C} by using Equation \eqref{eqn:v_ux_CABC_2}$_{m\to m+1}$, 
we obtain
\begin{subequations
\begin{align}
 &\widetilde{u} u+\dfrac{(p_l-q_mx)(q_{m+1}-p_l\widetilde{x})}{\widetilde{x}}=0,\\
 &\overline{u} u+\dfrac{(p_l-q_mx)(q_m-p_{l+1}\overline{x})}{x}=0,
\end{align}
\end{subequations}
respectively.
The equations above together with the P$\Delta$E s \eqref{eqn:dKdV} and \eqref{eqn:lsG} lead to the following theorem.

\begin{theorem}\label{thm:CAC1}
The following system has the CAC property but does not have the tetrahedron property:
\begin{subequations}\label{eqns:dKdV_lsG_CAC_1
\begin{align}
 &\widetilde{\ou}-u=\dfrac{\beta_{m+1}-\alpha_l}{\widetilde{u}}-\dfrac{\beta_m-\alpha_{l+1}}{\overline{u}},
 \label{eqn:dKdV_lsG_CAC_1_A}\\
 &\widetilde{u} u+\dfrac{(p_l-q_mx)(q_{m+1}-p_l\widetilde{x})}{\widetilde{x}}=0,
 \label{eqn:dKdV_lsG_CAC_1_B}\\
 &\overline{u} u+\dfrac{(p_l-q_mx)(q_m-p_{l+1}\overline{x})}{x}=0,
 \label{eqn:dKdV_lsG_CAC_1_C}\\
 &\dfrac{\,\widetilde{\ox}\,}{x}
 =\left(\dfrac{p_{l+1}-q_m\overline{x}}{q_m-p_{l+1}\overline{x}}\right)
 \left(\dfrac{q_{m+1}-p_l\widetilde{x}}{p_l-q_{m+1}\widetilde{x}}\right),
 \label{eqn:dKdV_lsG_CAC_1_AA}
\end{align}
\end{subequations}
where $p_l$ and $q_m$ are given by Equation \eqref{eqn:pq_albe_ga}.
Since the equations \eqref{eqn:dKdV_lsG_CAC_1_A} and \eqref{eqn:dKdV_lsG_CAC_1_AA} are equal to the P$\Delta$E s \eqref{eqn:dKdV} and \eqref{eqn:lsG}, respectively, we can also say that the pair of equations \eqref{eqn:dKdV_lsG_CAC_1_B} and \eqref{eqn:dKdV_lsG_CAC_1_C} is a CAC-type BT from the {P$\Delta$E} \eqref{eqn:dKdV} to the {P$\Delta$E} \eqref{eqn:lsG}.
\end{theorem}
\begin{proof}
Consider the cube in Figure \ref{fig:cube_ux}.
In the three different ways (see Appendix \ref{subsection:CAC_def}), $\widetilde{\ox}$ can be uniquely represented by the initial values $\{u,\overline{u},\widetilde{u},x\}$ as
\begin{equation
 \widetilde{\ox}=\dfrac{q_{m+1}\widetilde{u}\,\Big((\alpha_{l+1}-\beta_m)(p_l-q_mx)-q_m u\overline{u} x\Big)}{p_{l+1}\overline{u}\,\Big((\alpha_l-\beta_{m+1})(p_l-q_mx)-p_l u\widetilde{u}\Big)}.
\end{equation}
Since $\widetilde{\ox}$ depends on $u$, the tetrahedron property does not hold.
Therefore, we have completed the proof.
\end{proof}

Theorem \ref{thm:CAC1} gives the following corollary.

\begin{corollary}
Both of the P$\Delta$E s \eqref{eqn:dKdV} and \eqref{eqn:lsG} have the CAC property.
\end{corollary}

\begin{figure}[htbp]
\begin{center}
 \includegraphics[width=0.45\textwidth]{fig_cube_CAC_ux.eps}
\end{center}
\caption{
A cube for the CAC-systems \eqref{eqns:dKdV_lsG_CAC_1} and \eqref{eqns:dKdV_lsG_CAC_2}.
The $u$- and $x$-variables are assigned at the bottom vertices and the top vertices, respectively.
The corresponding equations of a CAC-system are assigned to the faces, e.g.
the equations \eqref{eqn:dKdV_lsG_CAC_1_A} and \eqref{eqn:dKdV_lsG_CAC_1_AA} are assigned to the bottom and top faces, respectively.
Also, as a convention, the parameters in a CAC-system are placed on the line segments.
Thus, the parameters $\{\alpha_l,\alpha_{l+1}\}$ are assigned to the line segments connecting the variables that represent a shift in the $l$-direction (e.g., $u$ and $\overline{u}$) since $\{\alpha_l,\alpha_{l+1}\}$ depend on only $l$.
Similarly, the parameters $\{\beta_m,\beta_{m+1}\}$ are assigned to the line segments connecting the variables that represent a shift in the $m$-direction.
On the other hand, the parameter $\gamma$ is assigned to the line segments connecting the $u$- and $x$-varibales (e.g., $u$ and $x$) since $\gamma$ is a parameter associated with the BT from the quad-equation given by $u$-variable to that given by $x$-variable.
For simplicity, only one parameter is shown for each in the figure, but they can also be placed on three other parallel line segments.
For example, $\gamma$ can be also placed on the line segment connecting $\overline{u}$ and $\overline{x}$.
}
\label{fig:cube_ux}
\end{figure}

\begin{remark}
If we choose 
\begin{equation
 v=\dfrac{p_lu(p_l-q_mx)}{{p_l}^2-{q_m}^2}
\end{equation}
as the solution of Equation \eqref{eqn:v_ux_CABC_1} instead of Equation \eqref{eqn:v_ux_CABC_2} and construct another system of P$\Delta$E s, then we can verify that the resulting system does not have CAC property.
\end{remark}

As shown in \cite{BS2002:MR1890049,HJN2016:MR3587455,NijhoffFW2002:MR1912127,WalkerAJ:thesis}, the Lax pair \eqref{eqn:Lax_phi} with \eqref{eqn:Lax_dKdV_CAC1} of the {P$\Delta$E} \eqref{eqn:dKdV} can be constructed from the equations \eqref{eqn:dKdV_lsG_CAC_1_B} and \eqref{eqn:dKdV_lsG_CAC_1_C} by setting
\begin{equation}
 x_{l,m}=\dfrac{F_{l,m}}{G_{l,m}},\quad
 \Phi_{l,m}=\begin{pmatrix}F_{l,m}\\G_{l,m}\end{pmatrix}.
\end{equation}
 

\begin{remark}
Similar to the Lax pair of the {P$\Delta$E} \eqref{eqn:dKdV}, 
we can also construct a Lax pair of the {P$\Delta$E} \eqref{eqn:lsG} from the equations \eqref{eqn:dKdV_lsG_CAC_1_B} and \eqref{eqn:dKdV_lsG_CAC_1_C} by setting 
\begin{equation}
 u_{l,m}=\dfrac{F_{l,m}}{G_{l,m}},\quad
 \Phi_{l,m}=\begin{pmatrix}F_{l,m}\\G_{l,m}\end{pmatrix}.
\end{equation}
However, in this case, we still have not figured out how to introduce a spectral parameter.
\end{remark}
 

\subsection{A CAC-type BT from the {P$\Delta$E} \eqref{eqn:dKdV} to the {P$\Delta$E} \eqref{eqn:lsG} (II)}\label{subsection:CABC_to_CAC_2}
In this subsection, using the CABC-type BTs \eqref{eqns:lsG_CABC} and \eqref{eqns:dKdV_CABC_2}, we obtain a CAC-type BT from the {P$\Delta$E} \eqref{eqn:dKdV} to the {P$\Delta$E} \eqref{eqn:lsG}.
Since the process for demonstrating the result is exactly the same as that for the CABC-type BTs \eqref{eqns:dKdV_CABC_1} and \eqref{eqns:lsG_CABC} discussed in \S \ref{subsection:CABC_to_CAC_1}, we omit detailed arguments.

\begin{theorem}\label{thm:CAC2}
The following system has the CAC property but does not have the tetrahedron property:
\begin{subequations}\label{eqns:dKdV_lsG_CAC_2
\begin{align}
 &\widetilde{\ou}-u=\dfrac{\beta_{m+1}-\alpha_l}{\widetilde{u}}-\dfrac{\beta_m-\alpha_{l+1}}{\overline{u}},
 \label{eqn:dKdV_lsG_CAC_2_A}\\
 &\widetilde{u} u+\dfrac{(q_m-p_lx)(p_l-q_{m+1}\widetilde{x})}{x}=0,
 \label{eqn:dKdV_lsG_CAC_2_B}\\
 &\overline{u} u+\dfrac{(q_m-p_lx)(p_{l+1}-q_m\overline{x})}{\overline{x}}=0,
 \label{eqn:dKdV_lsG_CAC_2_C}\\
 &\dfrac{\,\widetilde{\ox}\,}{x}
 =\left(\dfrac{p_{l+1}-q_m\overline{x}}{q_m-p_{l+1}\overline{x}}\right)
 \left(\dfrac{q_{m+1}-p_l\widetilde{x}}{p_l-q_{m+1}\widetilde{x}}\right),
 \label{eqn:dKdV_lsG_CAC_2_AA}
\end{align}
\end{subequations}
where $p_l$ and $q_m$ are given by Equation \eqref{eqn:pq_albe_ga}.
Since the equations \eqref{eqn:dKdV_lsG_CAC_2_A} and \eqref{eqn:dKdV_lsG_CAC_2_AA} are equal to the P$\Delta$E s \eqref{eqn:dKdV} and \eqref{eqn:lsG}, respectively, we can also say that the pair of equations \eqref{eqn:dKdV_lsG_CAC_2_B} and \eqref{eqn:dKdV_lsG_CAC_2_C} is a CAC-type BT from the {P$\Delta$E} \eqref{eqn:dKdV} to the {P$\Delta$E} \eqref{eqn:lsG}.
\end{theorem}
\begin{proof}
Consider the cube in Figure \ref{fig:cube_ux}.
In the three different ways (see Appendix \ref{subsection:CAC_def}), $\widetilde{\ox}$ can be uniquely represented by the initial values $\{u,\overline{u},\widetilde{u},x\}$ as
\begin{equation
 \widetilde{\ox}=\dfrac{p_{l+1}\overline{u}\,\Big((\alpha_l-\beta_{m+1})(q_m-p_lx)+p_l u\widetilde{u} x\Big)}{q_{m+1}\widetilde{u}\,\Big((\alpha_{l+1}-\beta_m)(q_m-p_lx)+q_m u\overline{u}\Big)}.
\end{equation}
Since $\widetilde{\ox}$ depends on $u$, the tetrahedron property does not hold.
Therefore, we have completed the proof.
\end{proof}

Setting
\begin{equation}
 x_{l,m}=\dfrac{F_{l,m}}{G_{l,m}},\quad
 \Phi_{l,m}=\begin{pmatrix}F_{l,m}\\G_{l,m}\end{pmatrix},
\end{equation}
from the equations \eqref{eqn:dKdV_lsG_CAC_2_B} and \eqref{eqn:dKdV_lsG_CAC_2_C}, we obtain the Lax pair \eqref{eqn:Lax_phi} with \eqref{eqn:Lax_dKdV_CAC2} of the {P$\Delta$E} \eqref{eqn:dKdV}.

\begin{remark}
Similar to the Lax pair of the {P$\Delta$E} \eqref{eqn:dKdV}, 
we can also construct a Lax pair of the {P$\Delta$E} \eqref{eqn:lsG} from the equations \eqref{eqn:dKdV_lsG_CAC_2_B} and \eqref{eqn:dKdV_lsG_CAC_2_C} by setting 
\begin{equation}
 u_{l,m}=\dfrac{F_{l,m}}{G_{l,m}},\quad
 \Phi_{l,m}=\begin{pmatrix}F_{l,m}\\G_{l,m}\end{pmatrix}.
\end{equation}
However, in this case, how to introduce a spectral parameter is still unsolved at this point.
\end{remark}

\subsection{A CAC-type auto-BT of the {P$\Delta$E} \eqref{eqn:lsG}}\label{subsection:2CACs_to_CAC}
In this subsection, using two CAC-systems \eqref{eqns:dKdV_lsG_CAC_1} and \eqref{eqns:dKdV_lsG_CAC_2}, we obtain a CAC-system, which implies a CAC-type auto-BT of the {P$\Delta$E} \eqref{eqn:lsG}.

Here, let us use the following CAC-system:
\begin{subequations}\label{eqns:dKdV_lsG_CAC_1_XPQ
\begin{align}
 &\widetilde{\ou}-u=\dfrac{\beta_{m+1}-\alpha_l}{\widetilde{u}}-\dfrac{\beta_m-\alpha_{l+1}}{\overline{u}},
 \label{eqn:dKdV_lsG_CAC_1_A_XPQ}\\
 &\widetilde{u} u+\dfrac{(P_l-Q_mX)(Q_{m+1}-P_l\widetilde{X})}{\widetilde{X}}=0,
 \label{eqn:dKdV_lsG_CAC_1_B_XPQ}\\
 &\overline{u} u+\dfrac{(P_l-Q_mX)(Q_m-P_{l+1}\overline{X})}{X}=0,
 \label{eqn:dKdV_lsG_CAC_1_C_XPQ}\\
 &\dfrac{\,\widetilde{\oX}\,}{X}
 =\left(\dfrac{P_{l+1}-Q_m\overline{X}}{Q_m-P_{l+1}\overline{X}}\right)
 \left(\dfrac{Q_{m+1}-P_l\widetilde{X}}{P_l-Q_{m+1}\widetilde{X}}\right),
 \label{eqn:dKdV_lsG_CAC_1_AA_XPQ}
\end{align}
\end{subequations}
where $X=X_{l,m}$ and 
\begin{equation}\label{eqn:PQ_albe_gala
 P_l=\sqrt{\alpha_l-\gamma-\lambda},\quad
 Q_m=\sqrt{\beta_m-\gamma-\lambda}\,,
\end{equation}
instead of the CAC-system \eqref{eqns:dKdV_lsG_CAC_1} with the variable $x_{l,m}$ and the parameters $p_l$, $q_m$ in the system \eqref{eqns:dKdV_lsG_CAC_1} replaced by the variable $X_{l,m}$ and the parameters $P_l$, $Q_m$, respectively.
Eliminating $\widetilde{u} u$ from the equations \eqref{eqn:dKdV_lsG_CAC_2_B} and \eqref{eqn:dKdV_lsG_CAC_1_B_XPQ}
and $\overline{u} u$ from the equations \eqref{eqn:dKdV_lsG_CAC_2_C} and \eqref{eqn:dKdV_lsG_CAC_1_C_XPQ}, we obtain
\begin{subequations}
\begin{align}
 &\dfrac{(q_m-p_lx)(p_l-q_{m+1}\widetilde{x})}{x}=\dfrac{(P_l-Q_mX)(Q_{m+1}-P_l\widetilde{X})}{\widetilde{X}},\\
 &\dfrac{(q_m-p_lx)(p_{l+1}-q_m\overline{x})}{\overline{x}}=\dfrac{(P_l-Q_mX)(Q_m-P_{l+1}\overline{X})}{X},
\end{align}
\end{subequations}
respectively.
The equations above together with the equations \eqref{eqn:dKdV_lsG_CAC_2_AA} and \eqref{eqn:dKdV_lsG_CAC_1_AA_XPQ} lead to the following theorem.

\begin{theorem}\label{thm:CAC3}
The following system has the CAC property but does not have the tetrahedron property:
\begin{subequations}\label{eqns:lsG_CAC
\begin{align}
 &\dfrac{\,\widetilde{\ox}\,}{x}
 =\left(\dfrac{p_{l+1}-q_m\overline{x}}{q_m-p_{l+1}\overline{x}}\right)
 \left(\dfrac{q_{m+1}-p_l\widetilde{x}}{p_l-q_{m+1}\widetilde{x}}\right),
 \label{eqn:lsG_CAC_A}\\
 &\dfrac{(q_m-p_lx)(p_l-q_{m+1}\widetilde{x})}{x}=\dfrac{(P_l-Q_mX)(Q_{m+1}-P_l\widetilde{X})}{\widetilde{X}},
 \label{eqn:lsG_CAC_B}\\
 &\dfrac{(q_m-p_lx)(p_{l+1}-q_m\overline{x})}{\overline{x}}=\dfrac{(P_l-Q_mX)(Q_m-P_{l+1}\overline{X})}{X},
 \label{eqn:lsG_CAC_C}\\
 &\dfrac{\,\widetilde{\oX}\,}{X}
 =\left(\dfrac{P_{l+1}-Q_m\overline{X}}{Q_m-P_{l+1}\overline{X}}\right)
 \left(\dfrac{Q_{m+1}-P_l\widetilde{X}}{P_l-Q_{m+1}\widetilde{X}}\right),
 \label{eqn:lsG_CAC_AA}
\end{align}
\end{subequations}
where
\begin{equation}\label{eqn:pq_PQ
 P_l=\sqrt{{p_l}^2-\lambda},\quad
 Q_m=\sqrt{{q_m}^2-\lambda}\,.
\end{equation}
Since both of the equations \eqref{eqn:lsG_CAC_A} and \eqref{eqn:lsG_CAC_AA} are equal to the {P$\Delta$E} \eqref{eqn:lsG}, we can also say that the pair of equations \eqref{eqn:lsG_CAC_B} and \eqref{eqn:lsG_CAC_C} is a CAC-type auto-BT of the {P$\Delta$E} \eqref{eqn:lsG}.
\end{theorem}
\begin{proof}
Consider the cube in Figure \ref{fig:cube_xX}.
In the three different ways (see Appendix \ref{subsection:CAC_def}), $\widetilde{\oX}$ can be uniquely represented by the initial values $\{x,\overline{x},\widetilde{x},X\}$ as
\begin{equation
 \widetilde{\oX}
 =\dfrac{Q_{m+1}(p_l-q_{m+1}\widetilde{x})\Big(({P_{l+1}}^2-{Q_m}^2)(P_l-Q_m X)\overline{x}+Q_m(p_{l+1}-q_m \overline{x})(q_m-p_lx)X\Big)}
 {P_{l+1}(p_{l+1}-q_m\overline{x})\Big(({P_l}^2-{Q_{m+1}}^2)(P_l-Q_m X)x+P_l(q_m-p_l x)(p_l-q_{m+1}\widetilde{x})\Big)}.
\end{equation}
Since $\widetilde{\oX}$ depends on $x$, the tetrahedron property does not hold.
Moreover, the condition \eqref{eqn:pq_PQ} can be obtained from the conditions \eqref{eqn:pq_albe_ga} and \eqref{eqn:PQ_albe_gala}.
Therefore, we have completed the proof.
\end{proof}

\begin{figure}[htbp]
\begin{center}
 \includegraphics[width=0.45\textwidth]{fig_cube_CAC_xX.eps}
\end{center}
\caption{
A cube for the CAC-system \eqref{eqns:lsG_CAC}.
See the caption of Figure \ref{fig:cube_ux} for the explanation of the placement of variables and parameters in the figure.
}
\label{fig:cube_xX}
\end{figure}

Setting
\begin{equation}
 X_{l,m}=\dfrac{F_{l,m}}{G_{l,m}},\quad
 \Psi_{l,m}=\begin{pmatrix}F_{l,m}\\G_{l,m}\end{pmatrix},
\end{equation}
from the equations \eqref{eqn:lsG_CAC_B} and \eqref{eqn:lsG_CAC_C}, we obtain the Lax pair \eqref{eqns:Lax_lsG} of the {P$\Delta$E} \eqref{eqn:lsG}.

\begin{remark}
Since Equation \eqref{eqn:lsG_CAC_AA} is also the lsG equation, setting
\begin{equation}
 x_{l,m}=\dfrac{F_{l,m}}{G_{l,m}},\quad
 \Psi_{l,m}=\begin{pmatrix}F_{l,m}\\G_{l,m}\end{pmatrix},
\end{equation}
we can construct another Lax pair of the lsG equation.
However, since the system \eqref{eqns:lsG_CAC} and the condition \eqref{eqn:pq_PQ} are invariant under the transformation
\begin{equation
 x_{l,m}\leftrightarrow \dfrac{1}{X_{l,m}},\quad
 p_l\leftrightarrow P_l,\quad
 q_m\leftrightarrow Q_m,\quad
 \lambda\leftrightarrow-\lambda,
\end{equation}
the resulting Lax pair is essentially the same as the Lax pair \eqref{eqns:Lax_lsG}.
\end{remark}

\section{Integrable systems on the 3-dimensional integer lattice}\label{section:3D}
In general, if a quad-equation has a CAC-type auto-BT, it can be easily extended to a system of P$\Delta$E s on the 3-dimensional integer lattice $\mathbb{Z}^3$. 
(See \cite{ABS2003:MR1962121,HJN2016:MR3587455,NW2001:MR1869690}.)
However, in the case of non-auto-BTs, a little ingenuity is required.
(See \cite{BollR2011:MR2846098}.)
In this section, we show how quad-equations that have CABC-type BTs, which are non-auto-BTs, can be extended to a system of P$\Delta$E s on the lattice $\mathbb{Z}^3$.
Moreover, we also show how the resulting system can be reduced to systems on the lattice $\mathbb{Z}^3$ associated with the CAC-type BTs.
\subsection{A system of P$\Delta$E s on the lattice $\mathbb{Z}^3$ associated with the CABC property}
In this subsection, using the three CABC-systems \eqref{eqns:dKdV_CABC_1}, \eqref{eqns:lsG_CABC} and \eqref{eqns:dKdV_CABC_2},
we construct an integrable system of P$\Delta$E s on the lattice $\mathbb{Z}^3$.
We also show the $\tau$-function of the resulting system.

\begin{remark}
Since Equation \eqref{eqn:dKdV_CABC_1_S} can be obtained from the equations \eqref{eqn:dKdV_CABC_1_B} and \eqref{eqn:dKdV_CABC_1_C}, we here omit Equation \eqref{eqn:dKdV_CABC_1_S} when using the CABC-system \eqref{eqns:dKdV_CABC_1} for simplicity.
For the same reason, the equations \eqref{eqn:lsG_CABC_S} and \eqref{eqn:dKdV_CABC_2_S} are also omitted when using the CABC-systems \eqref{eqns:lsG_CABC} and \eqref{eqns:dKdV_CABC_2}, respcetively.
\end{remark}

Let us denote the cubes for the three CABC-systems \eqref{eqns:dKdV_CABC_1}, \eqref{eqns:lsG_CABC} and \eqref{eqns:dKdV_CABC_2} by the symbols $C^{(1)}_{l,m}$, $C^{(2)}_{l,m}$ and $C^{(3)}_{l,m}$, respectively.
(See \S \ref{subsection:CABC_def} for the correspondence between a CABC-system and a cube.)
Overlap the cubes as in Figure \ref{fig_lattice_uvx} (Left), in the order $C^{(1)}_{l,m}$, $C^{(2)}_{l,m}$, $C^{(2)}_{l,m}$, $C^{(3)}_{l,m}$, so that the facing surfaces have the same equations.
We here also denote the resulting rectangular constructed by the set of four stacked cubes by the symbol $C^{(1223)}_{l,m}$.
Because both of the face-equations at the bottom and top of $C^{(1223)}_{l,m}$ are the dKdV equation, it can be placed repeatedly up and down as shown in Figure \ref{fig_lattice_uvx} (Right).
Note that since the parameter $\gamma$ is not included in the dKdV equation, a different $\gamma$ can be chosen for each stack of $C^{(1223)}_{l,m}$.
This repeated placement yields the following system of P$\Delta$E s on the lattice $\mathbb{Z}^3$:
\begin{subequations}\label{eqns:3D_uvx
\begin{align}
 &\widetilde{\ou}-u=\dfrac{\beta_{m+1}-\alpha_l}{\widetilde{u}}-\dfrac{\beta_m-\alpha_{l+1}}{\overline{u}},
 \label{eqn:3D_uvx_u}\\
 &\dfrac{\,\widetilde{\ox}\,}{x}=\left(\dfrac{p_{l+1,n}-q_{m,n}\overline{x}}{q_{m,n}-p_{l+1,n}\overline{x}}\right)\left(\dfrac{q_{m+1,n}-p_{l,n}\widetilde{x}}{p_{l,n}-q_{m+1,n}\widetilde{x}}\right),
 \label{eqn:3D_uvx_x}\\
 &(u-v)\left(\dfrac{\beta_m-\alpha_l}{u}+\widetilde{v}\right)-\beta_m+\gamma_n=0,
 \label{eqn:3D_uvx_uvB}\\
 &\dfrac{\beta_m-\alpha_l}{u}-\dfrac{\gamma_n-\alpha_l}{v}-\overline{u}+\overline{v}=0,
 \label{eqn:3D_uvx_uvC}\\
 &\dfrac{v\widetilde{v}}{{p_{l,n}}^2}-\dfrac{x(p_{l,n}-q_{m,n}x)}{p_{l,n}x-q_{m,n}}=0,
 \label{eqn:3D_uvx_xvB}\\
 &\dfrac{v\overline{v}}{p_{l,n}p_{l+1,n}}-\dfrac{\overline{x}(p_{l,n}-q_{m,n}x)}{p_{l,n}x-q_{m,n}}=0,
 \label{eqn:3D_uvx_xvC}\\
 &(\widehat{u}-\widetilde{v})\left(\dfrac{\beta_m-\alpha_l}{\widehat{u}}+v\right)-\beta_m+\gamma_n=0,
 \label{eqn:3D_uvx_TuvB}\\
 &\dfrac{\beta_m-\alpha_l}{\widehat{u}}-\dfrac{\gamma_n-\alpha_{l+1}}{\overline{v}}-\widehat{\ou}+v=0,
 \label{eqn:3D_uvx_TuvC}
\end{align}
\end{subequations}
where $u=u_{l,m,n}$, $v=v_{l,m,n}$, $x=x_{l,m,n}$ and 
\begin{equation
 p_{l,n}=\sqrt{\alpha_l-\gamma_n},\quad
 q_{m,n}=\sqrt{\beta_m-\gamma_n}.
\end{equation}

\begin{remark}
The correspondence between the cubes $C^{(1)}_{l,m}$, $C^{(2)}_{l,m}$, $C^{(3)}_{l,m}$ and the equations in the system \eqref{eqns:3D_uvx} is as follows.
{\rm
\begin{center}
\begin{tabular}{ll}
$C^{(1)}_{l,m}$ :&\eqref{eqn:3D_uvx_u}, \eqref{eqn:3D_uvx_uvB}, \eqref{eqn:3D_uvx_uvC}\\
$C^{(2)}_{l,m}$ :&\eqref{eqn:3D_uvx_x}, \eqref{eqn:3D_uvx_xvB}, \eqref{eqn:3D_uvx_xvC}\\
$C^{(3)}_{l,m}$ :&\eqref{eqn:3D_uvx_u}$_{n\to n+1}$, \eqref{eqn:3D_uvx_TuvB}, \eqref{eqn:3D_uvx_TuvC}
\end{tabular}
\end{center}
}
\end{remark}

\begin{figure}[htbp]
\begin{center}
 \includegraphics[width=0.4\textwidth]{fig_lattice_uvx_1.eps}\hspace{3em}
 \includegraphics[width=0.42\textwidth]{fig_lattice_uvx_2.eps}
\end{center}
\caption{
(Left): 
The rectangular $C^{(1223)}_{l,m}$ constructed by the four cubes.
Since the $u$-variable in the bottom cube $C^{(1)}_{l,m}$ and that in the top cube $C^{(3)}_{l,m}$ can be treated as different, we used the symbol $U$ for the $u$-variable of $C^{(3)}_{l,m}$ to differentiate them.\\
(Right): 
Two rectangulars $C^{(1223)}_{l,m}$.
Since we choose a different parameter $\gamma$ for each $C^{(1223)}_{l,m}$, we place the parameters $\gamma_n$ on the vertical line segments of the rectangles.
}
\label{fig_lattice_uvx}
\end{figure}

The $\tau$-function is one of the most important objects in the theory of integrable systems.
We show the $\tau$-function of the system \eqref{eqns:3D_uvx} in the following theorem.

\begin{theorem}\label{theorem:tau}
Let $\tau=\tau_{l,m,n}$ be the solution of the following bilinear equations:
\begin{subequations}\label{eqns:3D_uvx_bilinear
\begin{align}
 &p_{l,n}\widehat{\wtau}\,\overline{\tau}-q_{m,n}\widetilde{\tau}\,\widehat{\otau}+\tau\,\widehat{\wotau}=0,\label{eqn:3D_uvx_bilinear_1}\\
 &q_{m,n}\widehat{\wtau}\,\overline{\tau}-p_{l,n}\widetilde{\tau}\,\widehat{\otau}-\widehat{\tau}\,\widetilde{\otau}=0.\label{eqn:3D_uvx_bilinear_2}
\end{align}
\end{subequations}
Then, the $\tau$-function of the system \eqref{eqns:3D_uvx} is given as
\begin{equation}\label{eqn:3D_uvx_tau
 u=-\dfrac{\,\tau\,\widetilde{\otau}\,}{\overline{\tau}\,\widetilde{\tau}},\quad
 v=p_{l,n}\,\dfrac{\,\tau\,\widehat{\otau}\,}{\overline{\tau}\,\widehat{\tau}},\quad
 x=\dfrac{\,\widehat{\otau}\,\widetilde{\tau}\,}{\overline{\tau}\,\widehat{\wtau}}.
\end{equation}
\end{theorem}
\begin{proof}
The proof is given in Appendix \ref{section:proof_tau}.
\end{proof}

\subsection{Systems of P$\Delta$E s on the lattice $\mathbb{Z}^3$ associated with the CAC property}
In this subsection, we derive integrable systems on the lattice $\mathbb{Z}^3$ associated with the CAC property from the system \eqref{eqns:3D_uvx}.

As in \S \ref{subsection:CABC_to_CAC_1} and \S \ref{subsection:CABC_to_CAC_2}, eliminating the $v$-variable from the system \eqref{eqns:3D_uvx} we obtain the following system of P$\Delta$E s depending only on the $u$- and $x$-variables:
\begin{subequations}\label{eqns:3D_ux
\begin{align}
 &\widetilde{\ou}-u=\dfrac{\beta_{m+1}-\alpha_l}{\widetilde{u}}-\dfrac{\beta_m-\alpha_{l+1}}{\overline{u}},
 \label{eqn:3D_ux_u}\\
 &\dfrac{\,\widetilde{\ox}\,}{x}=\left(\dfrac{p_{l+1,n}-q_{m,n}\overline{x}}{q_{m,n}-p_{l+1,n}\overline{x}}\right)\left(\dfrac{q_{m+1,n}-p_{l,n}\widetilde{x}}{p_{l,n}-q_{m+1,n}\widetilde{x}}\right),
 \label{eqn:3D_ux_x}\\
 &\widetilde{u}\,u+\dfrac{(p_{l,n}-q_{m,n}x)(q_{m+1,n}-p_{l,n}\widetilde{x})}{\widetilde{x}}=0,
 \label{eqn:3D_ux_B1}\\
 &\overline{u}\,u+\dfrac{(p_{l,n}-q_{m,n}x)(q_{m,n}-p_{l+1,n}\overline{x})}{x}=0,
 \label{eqn:3D_ux_C1}\\
 &\widehat{\wu}~\widehat{u}+\dfrac{(q_{m,n}-p_{l,n}x)(p_{l,n}-q_{m+1,n}\widetilde{x})}{x}=0,
 \label{eqn:3D_ux_B2}\\
 &\widehat{\ou}~\widehat{u}+\dfrac{(q_{m,n}-p_{l,n}x)(p_{l+1,n}-q_{m,n}\overline{x})}{\overline{x}}=0.
 \label{eqn:3D_ux_C2}
\end{align}
\end{subequations}
Indeed, eliminating $\widetilde{v}$ from the equations \eqref{eqn:3D_uvx_uvB} and \eqref{eqn:3D_uvx_xvB},
we obtain
\begin{equation
 v=\dfrac{p_{l,n}ux}{p_{l,n}x-q_{m,n}}.
\end{equation}
Then, eliminating $v$-variable from the equations \eqref{eqn:3D_uvx_uvB} and \eqref{eqn:3D_uvx_uvC} by using the equation above, we obtain the equations \eqref{eqn:3D_ux_B1} and \eqref{eqn:3D_ux_C1}, respectively.
Moreover, eliminating $\widetilde{v}$ from the equations \eqref{eqn:3D_uvx_xvB} and \eqref{eqn:3D_uvx_TuvB},
we obtain
\begin{equation
 v=\dfrac{p_{l,n}(p_{l,n}-q_{m,n}x)}{\widehat{u}},
\end{equation}
and then eliminating $v$-variable from the equations \eqref{eqn:3D_uvx_TuvB} and \eqref{eqn:3D_uvx_TuvC} by using the equation above, we obtain the equations \eqref{eqn:3D_ux_B2} and \eqref{eqn:3D_ux_C2}, respectively.

\begin{remark}
As shown in Figure \ref{fig:lattice_ux_x} (Left), the system \eqref{eqns:3D_ux} can be thought of as a superposition of the CAC-systems \eqref{eqns:dKdV_lsG_CAC_1} and \eqref{eqns:dKdV_lsG_CAC_2} alternating in the $n$-direction.
\end{remark}

\begin{figure}[htbp]
\begin{center}
 \includegraphics[width=0.45\textwidth]{fig_lattice_ux.eps}\hspace{2em}
 \includegraphics[width=0.45\textwidth]{fig_lattice_x.eps}
\end{center}
\caption{
(Left): 
Cubes for the CAC-system \eqref{eqns:3D_ux}.
The system of equations \eqref{eqn:3D_ux_u}, \eqref{eqn:3D_ux_x}, \eqref{eqn:3D_ux_B1} and \eqref{eqn:3D_ux_C1} correspond to the lower cube, 
and the system of equations \eqref{eqn:3D_ux_u}$_{n\to n+1}$, \eqref{eqn:3D_ux_x}, \eqref{eqn:3D_ux_B2} and \eqref{eqn:3D_ux_C2} correspond to the upper cube.\\
(Right): 
A cube for the CAC-system \eqref{eqns:3D_x}.
}
\label{fig:lattice_ux_x}
\end{figure}

As in \S \ref{subsection:2CACs_to_CAC}, 
we eliminate $\widehat{\wu}\,\widehat{u}$ from the equations \eqref{eqn:3D_ux_B1}$_{n\to n+1}$ and \eqref{eqn:3D_ux_B2}.
Also, we eliminate $\widehat{\ou}\,\widehat{u}$ from the equations \eqref{eqn:3D_ux_C1}$_{n\to n+1}$ and \eqref{eqn:3D_ux_C2}.
The resulting equations together with Equation \eqref{eqn:3D_ux_x} lead to the following system of P$\Delta$E s depending only on the $x$-variable:
\begin{subequations}\label{eqns:3D_x
\begin{align}
 &\dfrac{\,\widetilde{\ox}\,}{x}=\left(\dfrac{p_{l+1,n}-q_{m,n}\overline{x}}{q_{m,n}-p_{l+1,n}\overline{x}}\right)\left(\dfrac{q_{m+1,n}-p_{l,n}\widetilde{x}}{p_{l,n}-q_{m+1,n}\widetilde{x}}\right),
 \label{eqn:3D_x_x}\\
 &\dfrac{(q_{m,n}-p_{l,n}x)(p_{l,n}-q_{m+1,n}\widetilde{x})}{x}
 =\dfrac{(p_{l,n+1}-q_{m,n+1}\widehat{x})(q_{m+1,n+1}-p_{l,n+1}\widehat{\wx})}{\widehat{\wx}},
 \label{eqn:3D_x_B}\\
 &\dfrac{(q_{m,n}-p_{l,n}x)(p_{l+1,n}-q_{m,n}\overline{x})}{\overline{x}}
 =\dfrac{(p_{l,n+1}-q_{m,n+1}\widehat{x})(q_{m,n+1}-p_{l+1,n+1}\widehat{\ox})}{\widehat{x}}.
 \label{eqn:3D_x_C}
\end{align}
\end{subequations}

\vspace{0.5em}
\begin{remark}
~\\[-1em]
\begin{enumerate}
\item 
As shown in Figure \ref{fig:lattice_ux_x} (Right), the system \eqref{eqns:3D_x} can be thought of as a superposition of the CAC-system \eqref{eqns:lsG_CAC} in the $n$-direction.
\item
Both equations \eqref{eqn:3D_x_B} and \eqref{eqn:3D_x_C} are also the lsG equation.
Indeed, by setting
\begin{equation
 y_{m,n}=\dfrac{q_{m,n}-p_{l,n}x_{l,m,n}}{\sqrt{\alpha_l-\beta_m}\,x_{l,m,n}},\quad
 a_m=-\sqrt{\alpha_l-\beta_m},\quad
 b_n=p_{l,n},
\end{equation}
Equation \eqref{eqn:3D_x_B} can be rewritten as the following standard form of the lsG equation in $(m,n)$-direction:
\begin{equation
 \dfrac{y_{m+1,n+1}}{y_{m,n}}
 =\left(\dfrac{b_{n+1}-a_my_{m,n+1}}{a_m-b_{n+1}y_{m,n+1}}\right)
 \left(\dfrac{a_{m+1}-b_ny_{m+1,n}}{b_n-a_{m+1}y_{m+1,n}}\right),
\end{equation}
while by setting
\begin{equation
 z_{l,n}=\dfrac{p_{l,n}x_{l,m,n}-q_{m,n}}{\sqrt{\beta_m-\alpha_l}},\quad
 c_l=-\sqrt{\beta_m-\alpha_l},\quad
 d_n=q_{m,n},
\end{equation}
Equation \eqref{eqn:3D_x_C} can be also rewritten as the following standard form of the lsG equation in $(l,n)$-direction:
\begin{equation
 \dfrac{z_{l+1,n+1}}{z_{l,n}}
 =\left(\dfrac{d_{n+1}-c_lz_{l,n+1}}{c_l-d_{n+1}z_{l,n+1}}\right)
 \left(\dfrac{c_{l+1}-d_nz_{l+1,n}}{d_n-c_{l+1}z_{l+1,n}}\right).
\end{equation}
\end{enumerate}
\end{remark}

\section{Concluding remarks}\label{ConcludingRemarks}
In this paper, using the CABC properties of the dKdV equation \eqref{eqn:dKdV} and the lsG equation \eqref{eqn:lsG}, we have shown that they have the CAC property.
Using the CABC and CAC properties, we also obtained their new Lax pairs.
Moreover, we have constructed the integrable system \eqref{eqns:3D_uvx} on the lattice $\mathbb{Z}^3$ by stacking the cubes associated with the CABC property in the third direction and have clarified the structure of its $\tau$-function.

In addition to the dKdV and lsG equations, there exist other integrable P$\Delta$E s for which we still do not know whether they have the CAC property or not.
We have shown that it is useful to consider the CABC property to clarify the CAC property of P$\Delta$E s.
Many studies have been done to find P$\Delta$E s which have the CAC property, e.g. \cite{ABS2003:MR1962121,ABS2009:MR2503862,BollR2011:MR2846098,BollR2012:MR3010833,HietarintaJ2004:MR2045902,HietarintaJ2005:MR2217106,HietarintaJ2019:zbMATH07053246}.
However, the dKdV and lsG equations have the CAC property but are not included in these previous studies.
For the reasons above, classifying P$\Delta$E s with the CABC property may be important for clarifying the CAC property of known integrable P$\Delta$E s. 
Moreover, it may lead to finding new integrable P$\Delta$E s. 
This should be a subject of future research.
\subsection*{Acknowledgment}
The author would like to thank Dr. Pavlos Kassotakis and Dr. Yang Shi for fruitful discussions.
This research was supported by a JSPS KAKENHI Grant Number JP19K14559.
