A discrete integrability test based on multiscale analysis]
A discrete integrability test based
on multiscale analysis

R. HERNÁNDEZ HEREDERO, D. LEVI and C. SCIMITERNA]
R. HERNÁNDEZ HEREDERO${}^\dag$, D. LEVI${}^\diamond$ and C. SCIMITERNA${}^{\diamond}$

Departamento de Matemática Aplicada
Universidad Politécnica de Madrid,
Escuela Universitaria de Ingeniería Técnica de Telecomunicación,
Campus Sur, Ctra. de Valencia Km. 7, 28031 Madrid, Spain

${}^\diamond$Dipartimento di Matematica e Fisica,
Università degli Studi Roma Tre and Sezione INFN, Roma Tre,
Via della Vasca Navale 84, 00146 Roma, Italy

In this article we present the results obtained applying the multiple scale expansion up to the order $\ep^6$ to a dispersive multilinear class of equations on a square lattice depending on 13 parameters. We show that the integrability conditions given by the multiple scale expansion give rise to 4 nonlinear equations, 3 of which are new, depending at most on 2 parameters and containing integrable sub cases. Moreover at least one sub case provides an example of a new integrable system.

§ INTRODUCTION

Discrete equations play an important role in Mathematical Physics for its double role. From one side discrete space time seems to be basic in the description of fundamental phenomena of nature as provided by quantum gravity. From the other, from discrete equations one can easily by continuous limit obtain differential difference and differential equations and thus discrete equations may provide good numerical schemes for integrating differential equations. A classification of integrable partial difference equation has been given by Adler, Bobenko and Suris [2] in the particular case of equations defined on four lattice points using the consistency around the cube condition with some symmetry constrains to be able to get definite results. Due to the constraints introduced, this classification is partial and already new equations with respect to those contained in the ABS classification have been obtained [16, 14, 10, 7, 6, 1].

In this paper we provide necessary conditions for the integrability of a class of real, autonomous difference equations in the variable $u: \mathbb Z^2
\rightarrow \mathbb R$  defined on a $\mathbb{Z}^2$ square-lattice

(u_n,m,u_n+ 1,m,u_n,m + 1, u_n + 1,m + 1; β_1, β_2,...)=0,
where the $\beta_i$'s are real, autonomous parameters. Integrability conditions will be determined through a  multiscale perturbative expansion. This approach has the distinctive advantage of providing criteria in a manner completely independent from other current approaches. Multiscale developments can be used to reinforce, enhance or augment our previous knowledge of discrete integrable systems given by other techniques.

To be able to propagate in all the $\mathbb Z^2$ plain, we will suppose, as in [2], that (<ref>) is linear-affine in every variable, implying that the equation is invariant under
the  Möbius transformation $T$

\begin{equation}\label{eqMob}
u_{n,m}\overset{T}{\mapsto} u_{n,m}'=\frac{Au_{n,m}+B}{Cu_{n,m}+D}.
\end{equation}

thus providing a geometrical significance to the classification. In this case, (<ref>) reduces to a polynomial equation in its variables with at most fourth order nonlinearity:

_IV   =   f_0+a_00  u_00+ a_01  u_01 + a_10  u_10 + a_11  u_11+
(α_1-α_2) u_00  u_10 + (β_1-β_2) u_00  u_01

+ d_1 u_00  u_11+ d_2  u_01  u_10 +
(β_1+β_2)  u_10  u_11 +
(α_1+α_2)  u_01  u_11
(τ_1-τ_3) u_00  u_01  u_10+ (τ_1+τ_3) u_00  u_10  u_11
+ (τ_2+τ_4) u_00  u_01  u_11

(τ_2-τ_4)  u_10  u_01  u_11
+ f_1  u_00  u_01  u_10  u_11=0,
where all coefficients are taken to be real and independent on $n$ and $m$.
We consider here the multiple scale expansion around the dispersive solution

u_n,m = K^n Ω^m,
of the linearized equation of (<ref>). Rewriting the constants $K$ and $\Omega$ as $K=e^{\ri k}$ and $\Omega=e^{-\ri\omega}$, and introducing the solution (<ref>) into the linear part of
Eq. (<ref>) we get a dispersion relation $\omega=\omega\left(k\right)$

ω=arctan[ a_00 a_01 + a_10 a_11 +(a_00 a_11 + a_01 a_10 ) cos(k)/(a_00 a_11 - a_10 a_01) sin(k) ],
if $f_0=0$.
The solution (<ref>) of (<ref>) with $f_0=0$ is dispersive if $\omega(k)$ is a real nonlinear function of the wave number $k$. This leads to the  constraint

a_00^2 -a_01^2+a_10^2-a_11^2 +2(a_00 a_10-a_01 a_11) cos(k) = 0
The constraint (<ref>) implies that one of the two following conditions must be satisfied:

* $a_{00}=a_{11}\equiv a_1$, $a_{01}=a_{10}\equiv a_2$,

* $a_{00}=-a_{11}\equiv a_1$, $a_{01}=-a_{10}\equiv a_2$.

Then the dispersion relation (<ref>) reduces to:

ω_±(k) =arctan[±2a_1 a_2 ±(a_1^2+a_2^2)cos(k)/(a_1^2-a_2^2)sin(k)]
We denote the family of equations (<ref>) satisfying the condition (1) with dispersion relation $\omega_{+}(k)$ as $\CQ^+$ and the one with dispersion relation $\omega_{-}(k)$ as $\CQ^-$. In all the cases $a_1$ and  $a_2$ cannot be zero and their ratio cannot be equal to $\pm 1$ to get a nontrivial dispersion relation.

In the following we will consider the integrability conditions for the class of equations $\CQ^+$. The study of the class $\CQ^-$  is left to a future work.
The result of this work are a series of integrability theorems and a table  of equations, invariant under a restricted  Möbius transformations  that pass the very stringent integrability  conditions obtained by considering the multiple scale expansion up to $\ep^6$ order.

In Section <ref> we present the main result on the discrete multiscale integrability test and  all the conditions up to order $\ep^6$ for a dispersive discrete equation $\CQ$ defined on a square lattice which at the lowest order gives a Nonlinear Schrödinger Equation (NLSE)  and in Section <ref> we apply it to the classification of the dispersive multilinear equation  $\CQ^+$. Section <ref> is devoted to some conclusive remarks.

§ THE DISCRETE MULTISCALE INTEGRABILITY TEST

Let us consider a dispersive discrete equation of the form $\CQ$, which at the lowest perturbation order gives a NLSE. An example of such a case is given by $\CQ=\CQ^+$ however the results presented below will not be limited to such a case. In such a case the discrete multiscale integrability test  may be summarized as follows:

i. One considers a small amplitude solution of Eq. (<ref>) given by
$u_{n,m}=\ep w_{n,m}$, $0 < |\ep| \ll 1$. In such a way  (<ref>) will split into linear and nonlinear terms:
= ∑_i=1^N ^i _i=0,
where $N \in \mathbb{N}$ is the nonlinearity order.
$N$ will be infinite only if the nonlinearity of Eq. (<ref>) comes from a non-polynomial
function. In the case $\CQ=\CQ^+$ $N \le 4$.
In the formal expansion (<ref>) each term $\CQ_i$ contains only
homogeneous polynomials of degree $i$ in the field variables $w_{n,m}$ defined on the square.
If the discrete equation is dispersive then the linear part $\CQ_1$ admits a solution
w_{n,m}=\exp [\ri( \kappa n -\omega m)]=K^n \Omega^m,
$\omega=\omega(\kappa)$, the dispersion relation, is a real function of $\kappa$.

ii. The multiscale expansion of the basic field variable $w_{n,m}$ around the harmonic $K^n \Omega^m$reads
w_n,m= ∑_ℓ=0^∞ ^ℓ∑_α=-ℓ-1^ℓ+1
K^αn Ω^αm  u_ℓ+1^(α),

where $u^{(\alpha)}_\ell =u^{(\alpha)}_\ell (n_1, \{m_j\})$ is a bounded slowly varying function of its arguments and $u^{(-\alpha)}_{\ell}=\bar u^{(\alpha)}_{\ell}$,
$\bar u_{\ell}$ being  the complex conjugate of $u_{\ell}$ as we are looking at real solutions.
Here $n_1= \ep n$, $m_j = \ep^j m$ $j= 1, 2, \dots$  are the slow-varying lattice

iii. The nearest-neighbors fields are expanded according to the following
w_n + 1,m =  ∑_ℓ=0^∞ ^ℓ∑_α=-ℓ-1^ℓ+1
K^α(n+ 1) Ω^αm
∑_j= max (0, |α|-1)^ℓ
_ ℓ- j  u_j+1^(α) ,

w_n ,m + 1 =  ∑_ℓ=0^∞ ^ℓ∑_α=-ℓ-1^ℓ+1
K^αn Ω^α(m - 1)
∑_j= max (0, |α|-1)^ℓ
_ ℓ- j  u_j+1^(α) ,

w_n + 1 ,m + 1 =  ∑_ℓ=0^∞ ^ℓ∑_α=-ℓ-1^ℓ+1
K^α(n + 1) Ω^  α(m - 1)
∑_j= max (0, |α|-1)^ℓ
_ ℓ- j  u_j+1^(α) ,
where the operators $\CA_i,\CB_i,\CC_i,
are equal to one when $i=0$, while  for the lowest values of $i$ they are presented in the following Table:

& i=1 & i=2 & i=3& i=4\\ \hline\hline
& & & & \\
\CA_i  &  \delta_{n_1} & \frac12\delta_{n_1}^2 & \frac16\delta_{n_1}^3 & \frac{1}{24}\delta_{n_1}^4\\
& & & & \\ \hline
& & & & \\
\CB_i  &  \delta_{m_1} &\frac12\delta_{m_1}^2 + \delta_{m_2} &
\frac16\delta_{m_1}^3+\delta_{m_1}\delta_{m_2}+\delta_{m_3} & \frac{1}{24}\delta_{m_1}^4+\frac12\delta_{m_1}^2\delta_{m_2}+\frac12\delta_{m_2}^2+\delta_{m_1}\delta_{m_3}+\delta_{m_4}\\
& & & & \\ \hline
& & & & \\
\CC_i  & \nabla & \frac12\nabla^2+\delta_{m_2} &
\frac16 \nabla^3+\nabla\delta_{m_2}+\delta_{m_3} & \frac{1}{24}\nabla^4+\frac12\nabla^2\delta_{m_2}+\frac12\delta_{m_2}^2+\nabla\delta_{m_3}+\delta_{m_4}\\
& & & & \\ \hline
\end{array}

where  $\delta_{k}$ are the formal derivatives with respect to the index $k$, $\delta_k\doteq\partial_k$ and $\nabla\doteq\delta_{m_1}+\delta_{n_1}$. The operator $\delta_k$ can always be expressed in terms of powers of the difference operators by the well known identity

\delta_{k}= \sum_{i=1}^\infty
\frac{(-1)^{i-1}}{i}\Delta_{k}^i,

where $\Delta_{k}$ is the discrete first right difference
operator with respect to the variable $k$, i.e.
$\Delta_k u_k \doteq u_{k+1} - u_k$.

A function $f_k$  will be a slow-varying function of order $L$  if $\Delta_k^{L+1} f_k \approx 0$.  In such a case  the $\delta_k$-operators, which in principle are formal series containing infinite
powers of $\Delta_k$, when acting on slow-varying functions of finite
order $L$ reduce
to polynomials in $\Delta_k$ at most of order $L$.

We shall assume here  that we are dealing with functions of an infinite slow-varying order, i.e. $L=\infty$,
so that the $\delta_k$-operators
may be taken as differential operators acting on the indexes of the harmonics $u_j^{(\alpha)}$.

iv. When we substitute the expansions (<ref>-<ref>) into (<ref>), we get an equation of the following
∑_j ^j ∑_α _j^(α)K^αn Ω^αm =0,
i.e. we must have
\CW_j^{(\alpha)}=0$ for all $\alpha$ and $j$.
Let us notice that the equations $\CW_j^{(\alpha)}=0$ are equations for the slowly varying functions $u_{\ell+1}^{(\alpha)}$ with $\ell \leq j$.

The multiscale expansion of the $\CQ$ equation for functions of infinite order will thus give rise to a set of continuous partial differential equations. By assumption, at lowest order (slow-time $m_{2}$) we get a NLSE. To define the values of the constants appearing in $\CQ$ for which the equation is integrable, we will consider the orders beyond that at which one obtains for the first harmonic $u^{\left(1\right)}_{1}$ the (integrable) NLSE. The first attempts to go beyond the NLSE order in the case of partial difference equations have been presented by Santini, Degasperis and Manakov in [4] and by Kodama and Mikhailov using normal forms[9]. In [4], the authors, starting from integrable models, through a combination of  asymptotic functional analysis and spectral methods, succeeded in removing all the secular terms from the reduced equations, order by order. Their results are summarized in the following statements:

* The number of slow-time variables required for the amplitudes $u^{\left(\alpha\right)}_{j}$s  coincides with the number of nonvanishing coefficients $\omega_{j}\left(k\right)=\frac{1}{j!}\frac{d^j \omega(k)}{dk^j}$;

* The amplitude $u^{\left(1\right)}_{1}$ evolves at the slow-times $t_{\sigma}$, $\sigma\geq 3$ according to the $\sigma$-th equation of the NLSE hierarchy;

* The amplitudes of the higher perturbations of the first harmonic $u^{\left(1\right)}_{j}$, $j\geq 2$ evolve at the slow-times $t_{\sigma}$, $\sigma\geq 2$ according to certain linear, nonhomogeneous equations when taking into account some asymptotic boundary conditions.

From the previous statements one can conclude  that the cancellation at each stage of the perturbation process of all the secular terms  is a sufficient condition to uniquely fix the evolution equations followed by every $u^{\left(1\right)}_{j}$, $j\geq 1$ for each slow-time $t_\sigma$. Conversely from [5] we can affirm that this expansion results secularity-free. In this way this procedure provides necessary and sufficient conditions to get secularity-free reduced equations.

Following [5] we can state the following proposition:

If a nonlinear dispersive partial difference equation  is integrable, then under a multiscale expansion the functions $u^{\left(1\right)}_{l}$, $l\geq1$ satisfy the equations

\begin{eqnarray}
\partial_{t_{\sigma}}u^{\left(1\right)}_{1}=K_{\sigma}\left[u^{\left(1\right)}_{1}\right],\label{Valentia1}\ \ \ \ \ \ \ \ \ \ \ \ \ \\
M_{\sigma}u^{\left(1\right)}_{j}=f_{\sigma}(j),\ \ \ M_{\sigma}\doteq\partial_{t_{\sigma}}-K_{\sigma}^{\prime}\left[u^{\left(1\right)}_{1}\right],\label{Valentia2}
\end{eqnarray}

$\forall\ j,\ \sigma\geq 2$, where $K_{\sigma}\left[u^{\left(1\right)}_{1}\right]$ is the $\sigma$-th flow in the nonlinear Schrödinger hierarchy. All the other  $u_{j}^{(\kappa)}$, $\kappa\geq 2$ are expressed in terms of differential monomials of $u_{\rho}^{(1)}$, $\rho\leq j$.

In  (<ref>)   $f_{\sigma}(j)$ is a nonhomogeneous nonlinear forcing term depending on all the $u^{(1)}_{\kappa}$, $1\leq\kappa\leq j-1$, their complex conjugates and their $\xi$-derivatives, where $\xi$ is a variable depending on the group velocity and expressed through a linear combination of the slow-space and the first slow-time $t_{1}$, while $K_{\sigma}^{\prime}\left[u\right]v$ is the Frechet derivative of the nonlinear term $K_{\sigma}[u]$ along the direction $v$ defined by
K_{\sigma}^{\prime}[u]v\doteq\frac{d} {ds}K_{\sigma}[u+sv]\mid_{s=0},\nonumber
i.e. the linearization  of the expression $K_{\sigma}[u]$ along the direction $v$ near the function $u$.

In order to characterize the flows $K_{\sigma}\left[u^{\left(1\right)}_{1}\right]$ and the nonlinear forcing terms $f_{\sigma}(j)$, following [3], we introduce the finite dimensional vector spaces $\mathcal{P}_{\ell}$, $\ell\geq 2$, as being the set of all homogeneous, fully-nonlinear, differential polynomials in the functions $u_{j}^{(1)}$, $j\geq 1$, their complex conjugates and their $\xi$-derivatives of homogeneity degree $\ell$ in $\ep$ and $1$ in $e^{\ri\theta}$, where

\mbox{order}_{\ep}\left(\partial_{\xi}^{\kappa}u^{(1)}_{j}\right)=\mbox{order}_{\ep}\left(\partial_{\xi}^{\kappa}\bar u^{(1)}_{j}\right)=\kappa+j,\quad \kappa\geq 0.

We introduce the subspaces $\mathcal{P}_{ \ell}(\jmath)$ of $\mathcal{P}_{\ell}$, $\jmath\geq 1$, $\ell\geq 2$, whose elements are homogeneous, fully-nonlinear, differential polynomials in the functions $u_{k}^{(1)}$, their complex conjugates and their $\xi$-derivatives, with $1\leq k\leq \jmath$. From these definitions it follows that $\mathcal{P}_{\ell}=\mathcal{P}_{\ell}\left(\ell-2\right)$, that is $\jmath\leq \ell-2$. In fact the terms $u^{(1)}_{\ell}$ and $\bar u^{(1)}_{\ell}$, as well as $\partial_{\xi}u^{(1)}_{\ell-1}$ and $\partial_{\xi}\bar u^{(1)}_{\ell-1}$, are not included in $\mathcal{P}_{\ell}$ as any monomial should enter nonlinearly and terms like $u^{(1)}_{\ell-1}$ and $\bar u^{(1)}_{\ell-1}$  cannot be combined with any other of the monomials $u^{(1)}_{1}$ or $\bar u^{(1)}_{1}$ to give the right homogeneity degree in $e^{\ri\theta}$. For the same reasons, terms of the types $\partial_{\xi}^{\kappa}u^{(1)}_{\ell-\kappa}$, $\partial_{\xi}^{\kappa}\bar u^{(1)!
}_{\ell-\kappa}$, $0\leq\kappa\leq \ell-1$ and $\partial_{\xi}^{\kappa}u^{(1)}_{\ell-\kappa-1}$, $\partial_{\xi}^{\kappa}\bar u^{(1)}_{\ell-\kappa-1}$, $0\leq\kappa\leq \ell-2$ cannot appear. So the space $\mathcal{P}_{\ell}(\jmath)$  is defined as that functional space generated by the base of monomials of the following types

\begin{eqnarray}
\prod_{\alpha,\beta,\gamma,\delta}\left(\partial_{\xi}^{\alpha}u^{(1)}_{\beta}\right)^{\rho\left(\alpha,\beta\right)}\left(\partial_{\xi}^{\gamma}\bar u^{(1)}_{\delta}\right)^{\sigma\left(\gamma,\delta\right)},\ \ \ \rho\left(\alpha,\beta\right)\geq 0,\ \ \forall\alpha,\beta,\ \ \ \sigma\left(\gamma,\delta\right)\geq 0,\ \ \forall\gamma,\delta,\nonumber
\end{eqnarray}

where the product is carried out for all $\alpha$, $\beta$, $\gamma$ and $\delta$ such that  $1\leq\beta,\delta\leq \jmath\leq \ell-2$, $0\leq\alpha\leq \ell-\beta-2$ and $0\leq\gamma\leq \ell-\delta-2$, so that

\begin{eqnarray}
\sum_{\alpha,\beta,\gamma,\delta}\left(\alpha+\beta\right)\rho\left(\alpha,\beta\right)+\left(\gamma+\delta\right)\sigma\left(\gamma,\delta\right)=\ell,\nonumber\\
\sum_{\alpha,\beta,\gamma,\delta}\rho\left(\alpha,\beta\right)-\sigma\left(\gamma,\delta\right)=1\nonumber.\ \ \ \ \ \ \ \ \ \ \
\end{eqnarray}

For $n\geq 3$ the subspaces $\mathcal{P}_{\ell}(\jmath)$, can be generated recursively starting from the lowest one, corresponding to $\ell=2$ by the following relation

\begin{eqnarray}
\mathcal{P}_{\ell}(\jmath)=\partial_{\xi}\mathcal{P}_{\ell-1}(\jmath)\cup\left\{\prod_{\beta,\delta}\left(u^{(1)}_{\beta}\right)^{\rho\left(\beta\right)}\left(\bar u^{(1)}_{\delta}\right)^{\sigma\left(\delta\right)}\right\},\nonumber
\end{eqnarray}

where $\rho\left(\beta\right)\geq 0$ $\forall\beta$, $\sigma\left(\delta\right)\geq 0$ $\forall\delta$ and the product is extended for $1\leq\beta,\delta\leq \jmath\leq \ell-2$, so that

\begin{eqnarray}
\sum_{\beta,\delta}\beta\rho\left(\beta\right)+\delta\sigma\left(\delta\right)=\ell,\ \ \ \sum_{\beta,\delta}\rho\left(\beta\right)-\sigma\left(\delta\right)=1.\nonumber
\end{eqnarray}

It is then clear that in general $K_{n}\left[u^{\left(1\right)}_{1}\right]\in\left\{\partial_{\xi}^{\ell}u^{\left(1\right)}_{1}\right\}\cup\mathcal{P}_{\ell+1}(1)$ and that $f_{\sigma}(j)\in\mathcal{P}_{\sigma+j}(j-1)$, $\forall\sigma$, $j\geq 2$.

Eqs. (<ref>) are necessary conditions for  integrability and represent a hierarchy of compatible evolutions for the function $u^{\left(1\right)}_{1}$ at different slow-times.  The compatibility of (<ref>)  implies some commutativity conditions among their r.h.s. $f_{\sigma}(j)$. If they are satisfied the operators $M_{\sigma}$ defined in Eq. (<ref>) commute among themselves.
Once we fix the index $j\geq 2$ in the set of Eqs. (<ref>), this commutativity condition implies the following compatibility conditions

\begin{eqnarray}
M_{\sigma}f_{\sigma'}\left(j\right)=M_{\sigma'}f_{\sigma}\left(j\right),\ \ \ \forall\, \sigma,\sigma'\geq 2,\label{Lavinia}
\end{eqnarray}

where, as $f_{\sigma}\left(j\right)$ and $f_{\sigma'}\left(j\right)$ are functions of the different perturbations of the fundamental harmonic up to degree $j-1$, the time derivatives $\partial_{t_{\sigma}}$, $\partial_{t_{\sigma'}}$ of those harmonics appearing respectively in $M_{\sigma}$ and $M_{\sigma'}$ have to be eliminated using the evolution equations (<ref>) up to the index $j-1$. The commutativity conditions (<ref>) turn out to be an integrability test.

We finally define the degree of integrability of a given equation:

If the relations (<ref>) are satisfied up to the index $j$, $j\geq 2$, we say that our equation is asymptotically integrable of degree $j$ or $A_{j}$-integrable.

Conjecturing that an $A_{\infty}$ degree of asymptotic integrability actually implies integrability, we have that under this assumption the relations (<ref>, <ref>) are a sufficient condition for the integrability or that integrability is a necessary condition to have a multiscale expansion where all the Eqs. (<ref>) are satisfied. So the multiscale integrability test tell us that $\CQ$ will be integrable if its multiscale expansion will follow all the infinite relations (<ref>, <ref>). The higher the degree of asymptotic integrability, the closer the equation will be to an integrable one. However, as we can test the conditions (<ref>, <ref>) only up to a finite order (actually $4$), from them we can only derive necessary conditions for integrability, so we will not be able to state with $100\%$ certainty that the discrete equation is integrable. The results obtained at a finite but sufficiently high o!
rder will have a good probability to correspond to an integrable equation, but we need to use other techniques to prove it with $100\%$ certainty.

Let us present for completeness in the following the lowest order conditions for asymptotic integrability of order $k$ or $A_{k}$-integrability conditions. To simplify the notation, we will use for $u^{\left(1\right)}_{j}$ the concise form $u(j)$, $j\geq 1$. Moreover, for  convenience of the reader, we list the fluxes $K_{\sigma}\left[u\right]$ of the NLSE hierarchy for $u$ up to $\sigma=5$:

\begin{align}
K_{1}[u]&\doteq Au_{\xi},\\
K_{2}[u]&\doteq-\ri\rho_{1}\left[u_{\xi\xi}+\frac{\rho_{2}} {\rho_{1}}|u|^2u \right], \label{Rutuli1}\\
K_{3}[u]&\doteq B\left[u_{\xi\xi\xi}+\frac{3\rho_{2}} {\rho_{1}}|u|^2u_{\xi}\right],\label{Rutuli2}\\
K_{4}[u]&\doteq-\ri C\left\{u_{\xi\xi\xi\xi}+\frac{\rho_{2}} {\rho_{1}}\left[\frac{3\rho_{2}} {2\rho_{1}}|u|^4u+4|u|^2u_{\xi\xi}+3u_{\xi}^2\bar u+2|u_{\xi}|^2u+u^2\bar u_{\xi\xi}\right]\right\},\label{Rutuli3}\\
K_{5}[u]&\doteq D\bigg\{u_{\xi\xi\xi\xi\xi}\nonumber\\
&\qquad{}+\frac{5\rho_{2}}{\rho_{1}}\left[\frac{3\rho_{2}}{2\rho_{1}}|u|^4u_{\xi}+|u_{\xi}|^2u_{\xi}+u\bar u_{\xi}u_{\xi\xi}+2\bar uu_{\xi}u_{\xi\xi}+uu_{\xi}\bar u_{\xi\xi}+|u|^2u_{\xi\xi\xi}\right]\bigg\},\label{Rutuli4}
\end{align}

and the corresponding $K_{\sigma}^{'}[u]v$ up to $\sigma=4$:

\begin{align}
&K_{1}^{\prime}[u]v= Av_{\xi},\label{Arenta}\\
&K_{2}^{\prime}[u]v=-\ri\rho_{1}\left\{v_{\xi\xi}+\frac{\rho_{2}} {\rho_{1}}\left[u^2\bar v+2|u|^2v\right]\right\},\label{ArtemideEfesina}\ \ \ \ \ \ \\
&K_{3}^{\prime}[u]v=B\left\{v_{\xi\xi\xi}+\frac{3\rho_{2}} {\rho_{1}}\left[|u|^2v_{\xi}+\bar uu_{\xi}v+uu_{\xi}\bar v\right]\right\},\label{Abruzzo3}\\
&K_{4}^{\prime}\left[u\right]v=-iC\left\{v_{\xi\xi\xi\xi}+\frac{\rho_{2}}{\rho_{1}}\left[u^{2}\bar v_{\xi\xi}+4|u|^{2}v_{\xi\xi}+2uu_{\xi}\bar v_{\xi}+2u\bar u_{\xi}v_{\xi}+6\bar u u_{\xi}v_{\xi}+4uu_{\xi\xi}\bar v+\right.\right.
\nonumber\\
&\qquad\quad\quad\qquad\left.\left.+3u_{\xi}^{2}\bar v+\frac{3\rho_{2}}{\rho_{1}}|u|^{2}u^{2}\bar v+4\bar u u_{\xi\xi}v+2u\bar u_{\xi\xi}v+\frac{9\rho_{2}}{2\rho_{1}}|u|^{4}v+2|u_{\xi}|^{2}v\right]\right\},
\end{align}

where $A\not=0$, $\rho_{1}\not=0$, $\rho_{2}$, $B\not=0$, $C\not=0$ and $D\not=0$, if $\rho_{2}\not=0$, are arbitrary real constants.

§.§ The $A_{1}$-integrability condition.

The $A_{1}$-integrability condition is given by the reality of the coefficient $\rho_{2}$ of the nonlinear term in the NLSE. It is obtained commuting the NLSE flux $K_{2}[u]$ with the flux $B\left[u_{\xi\xi\xi}+\tau |u|^2u_{\xi}+\mu u^2\bar u_{\xi}\right]$ with $\tau$ and $\mu$ constants. This commutativity condition gives, if $\rho_{2}\not =0$,

\begin{eqnarray}
\operatorname{Im}\left[\rho_{2}\right]=\operatorname{Im}\left[B\right]=\operatorname{Im}\left[\rho_{1}\right]=0,\ \ \ \ \ \tau=3\rho_{2}/\rho_{1},\ \ \ \ \ \mu=0.\label{Montesiepi}
\end{eqnarray}

We remark that, when $\rho_{2}\not=0$, by the same method it is possible to determine all the coefficients of all the higher NLSE-symmetries (<ref>) together with the reality conditions of the coefficients $A$, $C$ and $D$.

§.§ The $A_{2}$-integrability conditions.

The $A_{2}$- integrability conditions  are obtained choosing $j=2$ in the compatibility conditions (<ref>) with $\sigma=2$ and $\sigma'=3$ or alternatively $\sigma'=4$, respectively

\begin{gather}
\end{gather}

In this case  $f_{2}(2)$, $f_{3}(2)$ and $f_{4}(2)$ will be respectively identified by 2, ($a, b$), 5, ($\alpha, \beta, \gamma, \delta, \epsilon$), and 8, ($\theta_1, \cdots, \theta_8$), complex constants

\begin{align}
f_{2}(2)&\doteq au_{\xi}(1)|u(1)|^2+b\bar u_{\xi}(1)u(1)^2,\label{Abruzzo1}\\
f_{3}(2)&\doteq\alpha |u(1)|^4u(1)+\beta |u_{\xi}(1)|^2u(1)+\gamma u_{\xi}(1)^2\bar u(1)+\label{Abruzzo2}\\
&\qquad{}+\delta\bar u_{\xi\xi}(1)u(1)^2+\epsilon |u(1)|^2u_{\xi\xi}(1),\nonumber\\
f_{4}\left(2\right)&\doteq\theta_{1}|u\left(1\right)|^{4}u_{\xi}\left(1\right)+\theta_{2}|u\left(1\right)|^{2}u\left(1\right)^{2}\bar u_{\xi}\left(1\right)+\theta_{3}|u_{\xi}\left(1\right)|^{2}u_{\xi}\left(1\right)+\\
&\qquad{}+\theta_{4}u\left(1\right)\bar u_{\xi}\left(1\right)u_{\xi\xi}\left(1\right)+\theta_{5}\bar u\left(1\right)u_{\xi}\left(1\right)u_{\xi\xi}\left(1\right)+\theta_{6}u\left(1\right)u_{\xi}\left(1\right)\bar u_{\xi\xi}\left(1\right)+\nonumber\\
&\qquad{}+\theta_{7}|u\left(1\right)|^{2}u_{\xi\xi\xi}\left(1\right)+\theta_{8}u\left(1\right)^{2}\bar u_{\xi\xi\xi}\left(1\right).\nonumber
\end{align}

As $\rho_{2}\not=0$, eliminating from Eq. (<ref>) the derivatives of $u(1)$ with respect to the slow-times $t_{2}$ and $t_{3}$ using the evolutions (<ref>) with $\sigma=2$ and $\sigma'=3$ and equating term by term, we obtain the following 2 $A_{2}$-integrability conditions

\begin{eqnarray}
a=\bar a,\ \ \ b=\bar b.\label{CieloUrbico}
\end{eqnarray}

So we have two conditions obtained when requiring the reality of the coefficients $a$ and $b$. The expressions of $\alpha$, $\beta$, $\alpha$, $\delta$ in terms of $a$ and $b$ are:

\begin{eqnarray}
\alpha=\frac{3\ri Ba\rho_{2}} {4\rho_{1}^2},\ \ \ \beta=\frac{3\ri Bb} {\rho_{1}},\ \ \ \gamma=\frac{3\ri Ba} {2\rho_{1}},\ \ \ \delta=0,\ \ \ \epsilon=\gamma.\label{Molise}
\end{eqnarray}

The same integrability conditions (<ref>) can be derived using Eq. (<ref>). As in our analysis we will need them, here follow the explicit expressions of the coefficients of the forcing term $f_{4}\left(2\right)$

\begin{equation}
\begin{gathered}
\theta_{1}=\frac{6Ca\rho_{2}}{\rho_{1}^2},\ \ \ \theta_{2}=\frac{3Cb\rho_{2}}{\rho_{1}^2},\ \ \ \theta_{3}=\frac{\left(a+3b\right)C}{\rho_{1}},\ \ \ \theta_{4}=\frac{\left(a+4b\right)C}{\rho_{1}},
\\
\theta_{5}=\frac{5Ca}{\rho_{1}},\ \ \ \theta_{6}=\frac{\left(a+2b\right)C}{\rho_{1}},\ \ \ \theta_{7}=\frac{2Ca}{\rho_{1}},\ \ \ \theta_{8}=\frac{Cb}{\rho_{1}}.
\end{gathered}\label{Moly}
\end{equation}

§.§ The $A_{3}$-integrability conditions.

The $A_{3}$-integrability conditions are derived in a similar way setting $j=3$ in the compatibility conditions (<ref>) with $\sigma=2$ and $\sigma'=3$, so that $M_{2}f_{3}\left(3\right)=M_{3}f_{2}\left(3\right)$. In this case  $f_{2}(3)$ and $f_{3}(3)$ will be respectively identified by 12 and 26 complex constants

\begin{align}
f_{2}(3)&\doteq\tau_{1}|u(1)|^4u(1)+\tau_{2}|u_{\xi}(1)|^2u(1)+\tau_{3}|u(1)|^2u_{\xi\xi}(1)+\tau_{4}\bar u_{\xi\xi}(1)u(1)^2\nonumber\\
&\qquad{}+\tau_{7}\bar u_{\xi}(2)u(1)^2+\tau_{8}u(2)^2\bar u(1)+\tau_{9}|u(2)|^2u(1)+\tau_{10}u(2)u_{\xi}(1)\bar u(1)\label{Lazio4}\\
&\qquad{}+\tau_{11}u(2)\bar u_{\xi}(1)u(1)+\tau_{12}\bar u(2)u_{\xi}(1)u(1)+\tau_{5}u_{\xi}(1)^2\bar u(1)+\tau_{6}u_{\xi}(2)|u(1)|^2,\nonumber\\
f_{3}(3)&\doteq\gamma_{1}|u(1)|^4u_{\xi}(1)+\gamma_{2}|u(1)|^2u(1)^2\bar u_{\xi}(1)+\gamma_{3}|u(1)|^2u_{\xi\xi\xi}(1)\nonumber\\
&\qquad{}+\gamma_{5}|u_{\xi}(1)|^2u_{\xi}(1)+\gamma_{6}\bar u_{\xi\xi}(1)u_{\xi}(1)u(1)+\gamma_{7}u_{\xi\xi}(1)\bar u_{\xi}(1)u(1)\nonumber\\
&\qquad{}+\gamma_{9}|u(1)|^4u(2)+\gamma_{10}|u(1)|^2u(1)^2\bar u(2)+\gamma_{11}\bar u_{\xi}(1)u(2)^2+\gamma_{12}u_{\xi}(1)|u(2)|^2\nonumber\\
&\qquad{}+\gamma_{13}|u_{\xi}(1)|^2u(2)+\gamma_{14}|u(2)|^2u(2)+\gamma_{15}u_{\xi}(1)^{2}\bar u(2)+\gamma_{16}|u(1)|^2u_{\xi\xi}(2)\nonumber\\
&\qquad{}+\gamma_{17}u(1)^2\bar u_{\xi\xi}(2)+\gamma_{18}u(2)\bar u_{\xi\xi}(1)u(1)+\gamma_{19}u(2)u_{\xi\xi}(1)\bar u(1)\nonumber\\
&\qquad{}+\gamma_{21}u(2)u_{\xi}(2)\bar u(1)+\gamma_{22}\bar u(2)u_{\xi}(2)u(1)+\gamma_{23}u_{\xi}(2)u_{\xi}(1)\bar u(1)\nonumber\\
&\qquad{}+\gamma_{25}\bar u_{\xi}(2)u_{\xi}(1)u(1)+\gamma_{26}\bar u_{\xi}(2)u(2)u(1)+\gamma_{4}u(1)^2\bar u_{\xi\xi\xi}(1)  \nonumber \\
&\qquad{}+\gamma_{8}u_{\xi\xi}(1)u_{\xi}(1)\bar u(1)+\gamma_{20}\bar u(2)u_{\xi\xi}(1)u(1)+\gamma_{24}u_{\xi}(2)\bar u_{\xi}(1)u(1).\label{Lazio5}
\end{align}

Let us eliminate from Eq. (<ref>) with $j=3$ the derivatives of $u(1)$ with respect to the slow-times $t_{2}$ and $t_{3}$ using the evolutions (<ref>) respectively with $\sigma=2$ and $\sigma'=3$ and the derivatives of  $u(2)$ using the evolutions (<ref>) with $\sigma=2$ and $\sigma'=3$. Let us  equate the remaining terms term by term, if $\rho_{2}\not=0$,  and, indicating with $R_{i}$ and $I_{i}$ the real and imaginary parts of $\tau_{i}$, $i=1,\ldots,12$, we obtain the following 15 $A_{3}$-integrability conditions

\begin{equation}
\begin{gathered}
R_{1}=-\frac{aI_{6}} {4\rho_{1}},\qquad R_{3}=\frac{(b-a)I_{6}} {2\rho_{2}}-\frac{aI_{12}} {2\rho_{2}},\qquad R_{4}=\frac{R_{2}} {2}+\frac{(a-b)I_{6}} {4\rho_{2}}+\frac{aI_{12}} {4\rho_{2}},\\
R_{5}=\frac{R_{2}} {2}+\frac{(a-b)I_{6}} {4\rho_{2}}+\frac{(2b-a)I_{12}} {4\rho_{2}},\qquad R_{6}=-\frac{aI_{8}} {\rho_{2}},\qquad R_{7}=R_{12}+\frac{(a-b)I_{8}} {\rho_{2}},\\
R_{8}=R_{9}=0,\qquad R_{10}=R_{12},\qquad R_{11}=R_{12}+\frac{(a-2b)I_{8}} {\rho_{2}},\\
I_{4}=\frac{(b+a)R_{12}} {4\rho_{2}}+\frac{\rho_{1}I_{1}} {\rho_{2}}+\frac{I_{2}-I_{3}-2I_{5}} {4}+\frac{\left[2b(a-b)+a^2\right]I_{8}} {4\rho_{2}^2},\qquad I_{7}=0,\\
I_{9}=2I_{8},\qquad I_{10}=I_{12},\qquad I_{11}=I_{6}+I_{12}.
\end{gathered}\label{Siculi}
\end{equation}

For completeness we give the expressions of  $\gamma_{j}$, $j=1,\ldots,26$ as functions of $\tau_{i}$, $i=1,\ldots,12$:

\begin{equation}
\begin{gathered}
\gamma_{1}=\frac{3B} {8\rho_{1}^2}\left[-2bR_{12}-8\rho_{1}I_{1}+2(I_{2}-2I_{3}-2I_{5})\rho_{2}+\ri (b-5a)I_{6}+\frac{2a^2I_{8}} {\rho_{2}}-3\ri aI_{12}\right],
\\
\gamma_{2}=-\frac{3Ba} {4\rho_{1}^2}\left[\ri I_{6}+\frac{(a-2b)I_{8}} {\rho_{2}}+\tau_{12}\right],\quad \gamma_{3}=\frac{3\ri B\tau_{3}} {2\rho_{1}},\quad \gamma_{4}=0,\quad \gamma_{5}=\frac{3\ri B\tau_{2}} {2\rho_{1}},
\\
\gamma_{6}=\frac{3\ri B\tau_{4}} {\rho_{1}},\quad \gamma_{7}=\gamma_{5},\quad \gamma_{8}=\gamma_{3}+\frac{3\ri B\tau_{5}} {\rho_{1}},\quad \gamma_{9}=-\frac{3B(\rho_{2}I_{6}+3a\ri I_{8})} {4\rho_{1}^2},
\\
\gamma_{10}=\frac{3\ri B\rho_{2}R_{6}} {2\rho_{1}^2},\quad \gamma_{11}=0,\quad \gamma_{12}=\frac{3\ri B\tau_{9}} {2\rho_{1}},\quad \gamma_{13}=\frac{3\ri B\tau_{11}} {2\rho_{1}},\quad \gamma_{14}=0,
\\
\gamma_{15}=\frac{3\ri B\tau_{12}} {2\rho_{1}},\quad\gamma_{16}=\frac{3\ri B\tau_{6}} {2\rho_{1}},\quad \gamma_{17}=\gamma_{18}=0,\quad \gamma_{19}=\frac{3\ri B\tau_{10}} {2\rho_{1}},\quad \gamma_{20}=\gamma_{15},
\\
\gamma_{21}=\frac{3\ri B\tau_{8}} {\rho_{1}},\quad\gamma_{22}=\gamma_{12},\quad \gamma_{23}=\gamma_{16}+\gamma_{19},\quad \gamma_{24}=\gamma_{13},\quad \gamma_{25}=\frac{3\ri B\tau_{7}} {\rho_{1}},\quad \gamma_{26}=0.
\end{gathered}\label{gam}
\end{equation}

§.§ The $A_{4}$-integrability conditions.

The $A_{4}$-integrability conditions are derived similarly from (<ref>) with $j=4$, that is $M_{2}f_{3}\left(4\right)=M_{3}f_{2}\left(4\right)$. Now $f_{2}(4)$ and $f_{3}(4)$ are respectively defined by 34 and 77 complex constants

\begin{equation}\label{f24}
\begin{aligned}
f_{2}\left(4\right)&\doteq\eta_{1}|u(1)|^4u_{\xi}(1)+\eta_{2}|u(1)|^2u(1)^2\bar u_{\xi}(1)+\eta_{3}|u(1)|^2u_{\xi\xi\xi}(1)\\
&\qquad{}+\eta_{5}|u_{\xi}(1)|^2u_{\xi}(1)+\eta_{6}\bar u_{\xi\xi}(1)u_{\xi}(1)u(1)+\eta_{7}u_{\xi\xi}(1)\bar u_{\xi}(1)u(1)\\
&\qquad{}+\eta_{9}|u(1)|^4u(2)+\eta_{10}|u(1)|^2u(1)^2\bar u(2)+\eta_{11}\bar u_{\xi}(1)u(2)^2+\eta_{12}u_{\xi}(1)|u(2)|^2\\
&\qquad{}+\eta_{13}|u_{\xi}(1)|^2u(2)+\eta_{14}|u(2)|^2u(2)+\eta_{15}u_{\xi}(1)^{2}\bar u(2)+\eta_{16}|u(1)|^2u_{\xi\xi}(2)\\
&\qquad{}+\eta_{17}u(1)^2\bar u_{\xi\xi}(2)+\eta_{18}u(2)\bar u_{\xi\xi}(1)u(1)+\eta_{19}u(2)u_{\xi\xi}(1)\bar u(1)\\
&\qquad{}+\eta_{21}u(2)u_{\xi}(2)\bar u(1)+\eta_{22}\bar u(2)u_{\xi}(2)u(1)+\eta_{23}u_{\xi}(2)u_{\xi}(1)\bar u(1)\\
&\qquad{}+\eta_{25}\bar u_{\xi}(2)u_{\xi}(1)u(1)+\eta_{26}\bar u_{\xi}(2)u(2)u(1)+\eta_{4}u(1)^2\bar u_{\xi\xi\xi}(1)   \\
&\qquad{}+\eta_{8}u_{\xi\xi}(1)u_{\xi}(1)\bar u(1)+\eta_{20}\bar u(2)u_{\xi\xi}(1)u(1)+\eta_{24}u_{\xi}(2)\bar u_{\xi}(1)u(1)+\\
&\qquad{}+\eta_{27}u(1)\bar u_{\xi}(1)u(3)+\eta_{28}\bar u(1)u_{\xi}(1)u(3)+\eta_{29}u(1)u_{\xi}(1)\bar u(3)+\\
&\qquad{}+\eta_{30}u(1)\bar u(2)u(3)+\eta_{31}\bar u(1)u(2)u(3)+\eta_{32}u(1)u(2)\bar u(3)+\\
&\qquad{}+\eta_{33}|u(1)|^2u_{\xi}(3)+\eta_{34}u(1)^2\bar u_{\xi}(3),
\end{aligned}
\end{equation}

\begin{equation}
\begin{aligned}\label{f34}
f_{3}(4)&\doteq\kappa_{1}u(1)|u(1)|^6+\kappa_{2}|u(1)|^2\bar u(1)u_{\xi}(1)^2+\kappa_{3}|u(1)|^2u(1)|u_{\xi}(1)|^2+\kappa_{4}u(1)^3\bar u_{\xi}(1)^2\\
&\quad{}+\kappa_{5}|u(1)|^4u_{\xi\xi}(1)+\kappa_{6}|u(1)|^2u(1)^2\bar u_{\xi\xi}(1)+\kappa_{7}|u_{\xi}(1)|^2u_{\xi\xi}(1)+\kappa_{8}u_{\xi}(1)^2\bar u_{\xi\xi}(1)\\
&\quad{}+\kappa_{9}u(1)|u_{\xi\xi}(1)|^2+\kappa_{10}\bar u(1)u_{\xi\xi}(1)^2+\kappa_{11}\bar u(1)u_{\xi}(1)u_{\xi\xi\xi}(1)+\kappa_{12}u(1)\bar u_{\xi}(1)u_{\xi\xi\xi}(1)\\
&\quad{}+\kappa_{13}u(1)u_{\xi}(1)\bar u_{\xi\xi\xi}(1)+\kappa_{14}|u(1)|^2u_{\xi\xi\xi\xi}(1)+\kappa_{15}u(1)^2\bar u_{\xi\xi\xi\xi}(1)\\
&\quad{}+\kappa_{16}|u(1)|^2\bar u(1)u(2)^2+\kappa_{17}|u(1)|^2u(1)|u(2)|^2+\kappa_{18}u(1)^3\bar u(2)^2\\
&\quad{}+\kappa_{19}|u(1)|^2\bar u(1)u_{\xi}(1)u(2)+\kappa_{20}|u(1)|^2u(1)\bar u_{\xi}(1)u(2)+\kappa_{21}|u(1)|^2u(1)u_{\xi}(1)\bar u(2)\\
&\quad{}+\kappa_{22}u(1)^3\bar u_{\xi}(1)\bar u(2)+\kappa_{23}\bar u_{\xi}(1)u_{\xi\xi}(1)u(2)+\kappa_{24}u_{\xi}(1)\bar u_{\xi\xi}(1)u(2)\\
&\quad{}+\kappa_{25}u_{\xi}(1)u_{\xi\xi}(1)\bar u(2)+\kappa_{26}u(1)\bar u_{\xi\xi\xi}(1)u(2)+\kappa_{27}\bar u(1)u_{\xi\xi\xi}(1)u(2)\\
&\quad{}+\kappa_{28}u(1)u_{\xi\xi\xi}(1)\bar u(2)+\kappa_{29}\bar u_{\xi\xi}(1)u(2)^2+\kappa_{30}u_{\xi\xi}(1)|u(2)|^2+\kappa_{31}|u(1)|^4u_{\xi}(2)\\
&\quad{}+\kappa_{32}|u(1)|^2u(1)^2\bar u_{\xi}(2)+\kappa_{33}|u_{\xi}(1)|^2u_{\xi}(2)+\kappa_{34}u_{\xi}(1)^2\bar u_{\xi}(2)\\
&\quad{}+\kappa_{35}\bar u(1)u_{\xi\xi}(1)u_{\xi}(2)+\kappa_{36}u(1)\bar u_{\xi\xi}(1)u_{\xi}(2)+\kappa_{37}u(1)u_{\xi\xi}(1)\bar u_{\xi}(2)\\
&\quad{}+\kappa_{38}u(1)\bar u_{\xi}(1)u_{\xi\xi}(2)+\kappa_{39}\bar u(1)u_{\xi}(1)u_{\xi\xi}(2)+\kappa_{40}u(1)u_{\xi}(1)\bar u_{\xi\xi}(2)\\
&\quad{}+\kappa_{41}|u(1)|^2u_{\xi\xi\xi}(2)+\kappa_{42}u(1)^2\bar u_{\xi\xi\xi}(2)+\kappa_{43}\bar u_{\xi}(1)u(2)u_{\xi}(2)\\
&\quad{}+\kappa_{44}u_{\xi}(1)\bar u(2)u_{\xi}(2)+\kappa_{45}u_{\xi}(1)u(2)\bar u_{\xi}(2)+\kappa_{46}u(1)|u_{\xi}(2)|^2+\kappa_{47}\bar u(1)u_{\xi}(2)^2\\
&\quad{}+\kappa_{48}\bar u(1)u(2)u_{\xi\xi}(2)+\kappa_{49}u(1)\bar u(2)u_{\xi\xi}(2)+\kappa_{50}u(1)u(2)\bar u_{\xi\xi}(2)\\
&\quad{}+\kappa_{51}|u(2)|^2u_{\xi}(2)+\kappa_{52}u(2)^2\bar u_{\xi}(2)+\kappa_{53}|u(1)|^4u(3)+\kappa_{54}|u(1)|^2u(1)^2\bar u(3)\\
&\quad{}+\kappa_{55}\bar u(1)u(3)^2+\kappa_{56}u(1)|u(3)|^2+\kappa_{57}|u(2)|^2u(3)+\kappa_{58}u(2)^2\bar u(3)\\
&\quad{}+\kappa_{59}|u_{\xi}(1)|^2u(3)+\kappa_{60}u_{\xi}(1)^2\bar u(3)+\kappa_{61}u(1)\bar u_{\xi\xi}(1)u(3)+\kappa_{62}\bar u(1)u_{\xi\xi}(1)u(3)\\
&\quad{}+\kappa_{63}u(1)u_{\xi\xi}(1)\bar u(3)+\kappa_{64}u(1)\bar u_{\xi}(1)u_{\xi}(3)+\kappa_{65}\bar u(1)u_{\xi}(1)u_{\xi}(3)\\
&\quad{}+\kappa_{66}u(1)u_{\xi}(1)\bar u_{\xi}(3)+\kappa_{67}|u(1)|^2u_{\xi\xi}(3)+\kappa_{68}u(1)^2\bar u_{\xi\xi}(3)+\kappa_{69}u_{\xi}(1)\bar u(2)u(3)\\
&\quad{}+\kappa_{70}\bar u_{\xi}(1)u(2)u(3)+\kappa_{71}u_{\xi}(1)u(2)\bar u(3)+\kappa_{72}\bar u(1)u_{\xi}(2)u(3)+\kappa_{73}u(1)\bar u_{\xi}(2)u(3)\\
&\quad{}+\kappa_{74}u(1)u_{\xi}(2)\bar u(3)+\kappa_{75}u(1)\bar u(2)u_{\xi}(3)+\kappa_{76}\bar u(1)u(2)u_{\xi}(3)+\kappa_{77}u(1)u(2)\bar u_{\xi}(3).
\end{aligned}
\end{equation}

If we indicate with $S_{j}$ and $T_{j}$ respectively the real and imaginary parts of $\eta_{j}$, $j=1$, …, $34$, when $\rho_{2}\not=0$, the $A_{4}$-integrability conditions are represented by $48$ real relations whose expressions are presented in the  Appendix.

To study the distance between an integrable partial differential equation and its discretizations other integrability conditions have been constructed in [13], corresponding to $M_{4}f_{2}\left(3\right)=M_{2}f_{4}\left(3\right)$ ($A_{3}$-integrability conditions) and to $M_{4}f_{2}\left(5\right)=M_{2}f_{4}\left(5\right)$ ($A_{5}$-integrability conditions) in the subspaces characterized by  $u\left(2n\right)=0$, $n\geq1$ with purely imaginary coefficients. In this case the $A_{3}$-integrability conditions are given by one real relation which can be deduced from (<ref>) and corresponds to $I_{4}=\rho_{1}I_{1}/\rho_{2}+\left(I_{2}-I_{3}-2I_{5}\right)/4$. The next integrability condition, the $A_{5}$-integrability condition, is given by 14 real conditions, not included in the integrability conditions given here.

The results presented in this Section will be used in the following Sections to classify integrable nonlinear equation on the square lattice.

§ DISPERSIVE AFFINE-LINEAR EQUATIONS ON THE SQUARE LATTICE

The aim of this Section is to derive the necessary conditions for the integrability of  the simplest class of $\mathbb{Z}^2$-lattice equations, that of dispersive and multilinear equations (<ref>) defined on the square lattice, satisfying the condition (1) with dispersion relation $\omega_{+}(k)$, i.e.

\begin{equation}
\begin{aligned}
&\CQ^+\doteq a_1 (u_{n,m} + u_{n+1,m+1}) + a_2 (u_{n+1,m} + u_{n,m+1})  \\
& \qquad{}+  (\alpha_1-\alpha_2) \, u_{n,m}u_{n+1,m} +   (\alpha_1+\alpha_2)\,
u_{n,m+1}u_{n+1,m+1}  \\
&\qquad  + \,  (\beta_1-\beta_2)\, u_{n,m}u_{n,m+1} + (\beta_1+\beta_2)\,
u_{n+1,m}u_{n+1,m+1}  \\
&\qquad{}+ \, \gamma_1 u_{n,m}u_{n+1,m+1} + \gamma_2 u_{n+1,m}u_{n,m+1}  \\
&\qquad{}+ \, (\xi_1-\xi_3)\, u_{n,m}u_{n+1,m}u_{n,m+1} + (\xi_1+\xi_3)\,
u_{n,m}u_{n+1,m}u_{n+1,m+1}  \\
&\qquad{}+ \, (\xi_2-\xi_4)\, u_{n+1,m}u_{n,m+1}u_{n+1,m+1} + (\xi_2+\xi_4)\,
u_{n,m}u_{n,m+1}u_{n+1,m+1}  \\
&\qquad{}+ \, \zeta u_{n,m} u_{n+ 1,m} u_{n,m + 1} u_{n + 1,m + 1}=0, \label{q4}
\end{aligned}
\end{equation}

where $a_1,a_2 \in \mathbb R\setminus \{0\}$, $|a_1| \neq |a_2|$, are the coefficients appearing in the
linear part while  $\alpha_1,\alpha_2,\beta_1,\beta_2,$ $\gamma_1,\gamma_2,$
$\xi_1, \xi_2, \xi_3, \xi_4, \zeta$ are real parameters which enter in the nonlinear part of the system. Here we will look, by using
the multiscale procedure described in Section <ref> into the values of these coefficients such that the class $\CQ^+$ is $A_1$,  $A_2$, $A_3$ and $A_4$ integrable.

( 0,  0)(1,0)100
( 100,  0)(-1,0)100

( 0,100)(1,0)100
( 100,100)(-1,0)100

(  0, 0)(0,1)100
(  0, 100)(0,-1)100

(100, 0)(0,1)100
(100, 100)(0,-1)100

(0, 0)(1,1)100
(100, 0)(-1,1)100
(97, -3)$\bullet$
(-3, -3)$\bullet$
(-3, 97)$\bullet$
(97, 97)$\bullet$

Graphical representation of the quadratic nonlinearities of $\CQ^+$

To perform a classification of the equations $\CQ^+$, we need to find  the  set of transformations that leave them invariant, i.e. the equivalence group. As mentioned before, a generic multilinear equation of the form (<ref>)  is  invariant under a Möbius transformation (<ref>). The constant term $f_0$ and the differences $a_{00}-a_{11}$, $a_{01}-a_{10}$ transform according to

f_0T↦ f_0'=D^4f_0+
2 B^3D
(ξ_1+ξ_2)+ B^2 D^2

a_00-a_11T↦a_00'-a_11'=Δ[D^2 (a_00-a_11)+B^2
(ξ_1-ξ_2-ξ_3+ξ_4)-2 B D

a_01-a_10T↦ a_01'-a_10'=Δ[D^2 (a_01-a_10)-B^2
(ξ_1-ξ_2+ ξ_3-ξ_4)+2 B D
with $\Delta=A D-B C$. These formulas allow to determine when a given linear-affine equation (<ref>) can be transformed into one belonging to class $\CQ^+$. For this to happen all three terms must be null, so setting the l.h.s. of (<ref>) to zero we get three polynomial equations for $B/D$ or $D/B$. If simultaneously solved (over the reals), we have an equation of the class $\CQ^+$. One could try to write the conditions over the coefficients of a general linear-affine equation (<ref>) by using resultant calculations on the three polynomial conditions, but they turn out to be too complicated. A solution of  (<ref>)   is given by restricted simultaneous Möbius transformations $R$ of the form

u_n,m↦u_n,m'=u_n,m/(Cu_n,m+D),    ∀  n, m
which will be our equivalence transformation. Under  (<ref>) the coefficients of Eq.  (<ref>) undergo the following transformations:
a_1R↦ a_1'= D^3a_1,   a_2R↦ a_2'= D^3a_2,   α_1R↦α_1'= D^2 [α_1+C(a_1+a_2)
],   α_2R↦α_2'= D^2α_2,

β_1R↦ β_1'= D^2 [β_1+C(a_1+a_2)
],  β_2R↦β_2'= D^2β_2,

γ_1R↦ γ_1'= D^2 (γ_1+2C a_1 ),  γ_2R↦ γ_2'= D^2 (γ_2+2C a_2 ),

ξ_1R↦ ξ_1'= D ξ_1+12 C D [3C
+γ_1+γ_2+2 (α_1-α_2+β_1)],

ξ_2R↦ ξ_2'= D ξ_2+12 C D [3C
+γ_1+γ_2+2 (α_1+α_2+β_1)],

ξ_3R↦ ξ_3'= D ξ_3+12 C D [C(a_1
-a_2)+γ_1-γ_2+2 β_2],

ξ_4R↦ ξ_4'= D ξ_4+12 C D [C(a_1
-a_2)+γ_1-γ_2-2 β_2],

ζR↦ ζ'= ζ+C^2 [2C
+γ_1+γ_2+2 (α_1+β_1)]+2C
We will indicate by $\CN$ the number of free parameters (although not all of them essential under $R$) appearing in each sub case of q4. Its maximum number is $\CN=13$, the number of free coefficients in q4.

§.§ Classification at order $\ep^3$.

By performing the multiscale expansion of Eq. (<ref>), the following statement holds for the  $A_{1}$-asymptotic integrability

The lowest order necessary conditions for the integrability of equations $\CQ^+$ read:

* Case 1 ($\CN=9$):

α_2 = β_2 = 0,

ξ_1 = ξ_2,   ξ_3 = ξ_4.

* Case 2 ($\CN=7$) :

α_2 =  β_2,  α_1 = β_1,

a_1 =2 a_2 ,

γ_1 = 2 γ_2,

a_1 (ξ_1 - ξ_2) = - a_1 (ξ_3 - ξ_4) = -2α_2 γ_2.

* Case 3 ($\CN=7$):

α_2 = - β_2,  α_1 = β_1,

a_2 =2 a_1 ,

γ_2 = 2 γ_1,

a_1 (ξ_1 - ξ_2) = a_1 (ξ_3 - ξ_4)=
- α_2 γ_1.

* Case 4 ($\CN=8$):

a_2α_1 = a_2β_1=1/2 (a_1+

a_1 (ξ_1 - ξ_2) = -α_2 γ_1,

a_1 (ξ_3 - ξ_4) = β_2 γ_1 .

* Case 5 ($\CN=8$):

(a_2-a_1)β_2= (a_2+a_1) α_2,

2 a_1 a_2 (a_1 - a_2) α_1 = (a_1 + a_2) (γ_2 a_1^2-γ_1

2 a_1 a_2 β_1 = γ_1 a_2^2 + γ_2 a_1^2,

(a_2- a_1)(ξ_1 - ξ_2)=
(γ_1 - γ_2)α_2,

(a_2 - a_1)^2(ξ_3 - ξ_4)= [ γ_2 (a_2 - 3 a_1 )-
γ_1 (a_1 - 3 a_2 )] α_2  .

* Case 6 ($\CN=8$):

(a_2+a_1)β_2= (a_2- a_1) α_2,

2 a_1 a_2  α_1 = γ_1 a_2^2 + γ_2 a_1^2,

2 a_1 a_2 (a_1 - a_2) β_1 =(a_1 + a_2) (γ_2 a_1^2-γ_1

(a_2^2 - a_1^2)(ξ_1 - ξ_2)= [ γ_1 (a_1 - 3 a_2 )-
γ_2 (a_2 - 3 a_1 )] α_2,

(a_1+ a_2)(ξ_3 - ξ_4)= (γ_2 - γ_1)α_2.

The obtained six subclasses of equation q4 are invariant under the restricted  Möbius transformation (<ref>).

Proof: Following the procedure described in Section <ref> we
expand the fields appearing in equation $\CQ^+$ according to formulas
The lowest order necessary conditions for the integrability of $\CQ^+$ are obtained
by considering the equation $\CW_3$ (see (<ref>)), namely the order
$\ep^3$ of the multiscale expansion. At this order we get
the $m_2$-evolution equation for the harmonic $u_0^{(1)}$, that is a NLSE of the form
δ_m_2 u_1^(1) + ρ_1δ_ξ^2 u_1^(1)+
ρ_2 u_1^(1) |u_1^(1)|^2=0,     ξ≐n_1 - dω/d κ m_1,
where the coefficients $\rho_1$ and $\rho_2$ will depend on the parameters of
the equation $\CQ^+$ and on the wave parameters $\kappa$ and $\omega=\omega_+$, with
$\omega_+$ expressed in terms of $\kappa$ through the dispersion relation (<ref>).
According to our multiscale test the lowest order necessary
condition for $\CQ^+$ to be an integrable lattice equation is that Eq. (<ref>) be integrable itself, namely $\rho_1$ and $\rho_2$ have
to be real coefficients.

Let us outline the construction of Eq. (<ref>).
At $\CO(\ep)$ we get:

* for $\alpha=1$ a linear equation which is identically satisfied by the
dispersion relation (<ref>).

* for $\alpha=0$ a linear equation whose solution is $u_1^{(0)}=0$.

At $\CO(\ep^2)$, taking into account the dispersion relation (<ref>),
we get:

* for $\alpha=2$ an algebraic relation between $u_2^{(2)}$ and $u_1^{(1)}$.

* for $\alpha=1$ a linear wave equation for $u_1^{(1)}$, whose solution is
given by $u_1^{(1)}(n_1,m_1,m_2)=u_1^{(1)}(\xi,m_2)$, where
$\xi\doteq n_1 - (d\omega/d \kappa) m_1$.

* for $\alpha=0$ an algebraic relation between $u_2^{(0)}$ and $u_1^{(1)}$.

Notice that from the $\CO(\ep^2)$ we find that the dependence of all the harmonics
on the slow-variables $n_1$ and $m_1$ is given by $\xi$.

At $\CO(\ep^3)$, for $\alpha=1$, by using the results obtained at the previous
orders, one gets the NLSE (<ref>) with

\rho_1 = \frac{a_1 a_2 (a_1^2-a_2^2) \sin \kappa }{ (a_1^2+a_2^2+2 a_1 a_2
\cos \kappa)^2}, \qquad
\rho_2 = \CR_1 + \ri \CR_2, \label{r2}

_1=sinκ[ _1^(0) + _1^(1) cosκ+ _1^(2) cos^2 κ+ _1^(3) cos^3 κ+_1^(4) cos^4 κ]/(a_1+a_2)(a_1^2+a_2^2+2 a_1 a_2 cosκ)^2 [(a_1-a_2)^2 + 2 a_1 a_2 cosκ(1+ cosκ) ],

_2=_2^(0) + _2^(1) cosκ+ _2^(2) cos^2 κ+ _2^(3) cos^3 κ+_2^(4) cos^4
κ+_2^(5) cos^5 κ/(a_1+a_2)(a_1^2+a_2^2+2 a_1 a_2 cosκ)^2 [(a_1-a_2)^2 + 2 a_1 a_2 cosκ(1+ cosκ) ].
Here the coefficients $\CR_1^{(i)}$, $0 \leq i \leq 4$, and $\CR_2^{(i)}$, $0
\leq i \leq 5$, are polynomials
depending on the coefficients $a_1,a_2 ,\alpha_1,\alpha_2,\beta_1,\beta_2,$ $\gamma_1,\gamma_2,$
$\xi_1,...,\xi_4$ and their expressions are cumbersome, so that we omit

Note that $\rho_1$ is a real coefficient depending only on the parameters
of the linear part of $\CQ^+$, while $\rho_2$ is a complex one. Hence
the integrability of the NLSE (<ref>) is equivalent to the request
$\CR_2 =0 \; \forall \, \kappa$, that is
_2^(i)=0,     0≤i ≤5.
Eqs. (<ref>) are a nonlinear algebraic system of six equations in twelve unknowns.  By solving it  one gets the six solutions contained in Proposition 1.
These solutions are computed taking into account that
$a_1,a_2 \in \mathbb R\setminus \{0\}$ with $|a_1| \neq |a_2|$.
One can solve two of the  six equations (<ref>) for $\xi_1$
and $\xi_3$, thus expressing them in terms of the remaining ten coefficients.
The resulting system of four equations turns out to be
$\xi_2$ and $\xi_4$-independent and linear in the
four variables $\alpha_1$, $\beta_1$, $\gamma_1$ and $\gamma_2$. Therefore we may write the remaining four equations as
a matrix equation with coefficients depending nonlinearly on
$\alpha_2$, $\beta_2$, $a_1$ and $a_2$. The rank of the matrix is three. The six solutions are obtained by requiring that the matrix be of rank 3, 2, 1  and 0, and correspond to the six classes q4 that pass integrability conditions up to order $ \mathcal{O}(\varepsilon^3)$. A direct calculation proves the invariance of the six classes with respect to the restricted Möbius transformation $R$.

If the coefficients $a_1,a_2 ,\alpha_1,\alpha_2,\beta_1,\beta_2,$ $\gamma_1,\gamma_2,$
$\xi_1,...,\xi_4$ of equation $\CQ^+$ do not satisfy one of the conditions
given in (<ref>–<ref>) then $\CQ^+$ is not integrable.

Quadratic difference equations are a subclass of $\CQ^+$ which have attracted a deal of attention. These equations are not Möbius invariant, but we can spot those that belong to the class $\CQ^+$ and pass our integrability conditions, just by inspection of (<ref>–<ref>). We have:

For quadratic equations, when  $\xi_1=\xi_2=\xi_3=\xi_4=\zeta=0$ in equations $\CQ^+$, the
lowest order necessary conditions for the integrability of
the resulting equation read:

* Case Q1: $\alpha_2=\beta_2=0$;

* Case Q2: $\alpha_2=\beta_2$, $\alpha_1=\beta_1$, $a_1=2a_2$, $\gamma_1=\gamma_2=0$, $\alpha_{j}\not=0$, $j=1$, $2$;

* Case Q3: $\alpha_2=-\beta_2$, $\alpha_1=\beta_1$, $a_2=2a_1$, $\gamma_1=\gamma_2=0$, $\alpha_{j}\not=0$, $j=1$, $2$;

* Case Q4:  $\alpha_1=\beta_1=\gamma_1=\gamma_2=0$, $\left(\alpha_{2},\beta_{2}\right)\not=\left(0,0\right)$.

§.§ Classification at order $\ep^4$.

For what concerns the $A_{2}$-asymptotically integrable cases satisfying the integrability conditions (<ref>), the following statement holds

At order $\ep^4$, the necessary conditions for the integrability of equations $\CQ^+$ read:

* Case 1 ($\CN=9$):

α_2 = β_2 = 0,

ξ_1 = ξ_2,   ξ_3 = ξ_4.

* Case 4 ($\CN=8$):

a_1 (ξ_1 - ξ_2) = -α_2 γ_1,

a_1 (ξ_3 - ξ_4) = β_2 γ_1 ,

The corresponding two subclasses of equations are non overlapping and invariant under the restricted Möbius transformation (<ref>).

As one can see, of the six $A_{1}$-asymptotically integrable cases listed in Proposition <ref>, Case 1 and Case 4 automatically satisfy the $A_{2}$-integrability conditions (<ref>), while the remaining four cases 2, 3, 5 and 6 reduce to some sub cases.

For quadratic equations, when $\xi_1=\xi_2=\xi_3=\xi_4=\zeta=0$ in equations $\CQ^+$, the  order $\left(\ep^4\right)$ necessary conditions for the integrability of
the resulting equation read:

* Case Q1: $\alpha_2=\beta_2=0$;

* Case Q4:  $\alpha_1=\beta_1=\gamma_1=\gamma_2=0$, $\left(\alpha_{2},\beta_{2}\right)\not=\left(0,0\right)$.

As one can see, only two out the previous four quadratic cases in Remark 1 survive, the Cases Q1 and Q4: the first one is a sub case of Case 1, while the second is a sub case of Case 4.

§.§ Classification at order $\ep^5$.

It is possible to find all the cases satisfying the $A_{3}$-integrability conditions (<ref>). They are given by the following proposition

The necessary and sufficient conditions for $\ep^{5}$ asymptotic integrability are:

Case (a): ($\CN=4$)
α_2=β_2=0,   γ_2=α_1+β_1-γ_1,   a_2=2a_1,   (2α_1-3γ_1, 2β_1-3γ_1)≠(0, 0),

ξ_1=ξ_2=α_1β_1/2a_1,   ξ_3=ξ_4=-(α_1-γ_1)(β_1-γ_1)/2a_1,   ζ=γ_1[3γ_1^2-3γ_1(α_1+β_1)+4α_1β_1]/4a_1^2;

Case (b): ($\CN=4$)
α_2=β_2=0,   γ_1=α_1+β_1-γ_2,   a_1=2a_2,   (2α_1-3γ_2, 2β_1-3γ_2)≠(0, 0)

ξ_1=ξ_2=α_1β_1/2a_2,   ξ_3=ξ_4=(α_1-γ_2)(β_1-γ_2)/2a_2,   ζ=γ_2[3γ_2^2-3γ_2(α_1+β_1)+4α_1β_1]/4a_2^2;

Case (c): ($\CN=5$)
α_1=β_1=(a_1+a_2)γ_1/2a_1,   α_2=β_2=0,   γ_2=a_2γ_1/a_1,   ξ_1=ξ_2,

ξ_3=ξ_4=(a_2-a_1)γ_1^2/4a_1^2-(a_2-a_1)/(a_2+a_1)ξ_2,   ρ≐[8a_1^2ξ_2/(a_1+a_2)-3γ_1^2]1/(a_1+a_2)^2≠0;

Case (d): ($\CN=5$)
α_1=β_1=(a_1+a_2)γ_1/2a_1,   α_2=β_2=0,   γ_2=a_2γ_1/a_1,   ξ_1=ξ_2,

ξ_3=ξ_4=(a_1-a_2)γ_1^2/2a_1^2-(a_1-a_2)/(a_1+a_2)ξ_2,   ρ≐[8a_1^2ξ_2/(a_1+a_2)-3γ_1^2]1/(a_1+a_2)^2≠0;

Case (e): ($\CN=4$)
α_1=β_1=γ_1+γ_2/2,   α_2=β_2=0,   γ_2≠a_2γ_1/a_1,   a_2/a_1≠1/2, 2,

ξ_1=ξ_2=3(γ_1+γ_2)^2/8(a_1+a_2),   ξ_3=ξ_4=9(a_1-a_2)(a_1γ_2-a_2γ_1)^2/8a_1a_2(a_1+a_2)^2-a_1γ_2^2-a_2γ_1^2/8a_1a_2,

Notes: In all of the cases $a_{2}/a_{1}\not=(0$, $\pm 1)$; the values $a_{2}/a_{1}=(2, \,\frac12)$ are excluded in Case (e) because they would provide a sub case of Case (a) or of Case (b). All  the Cases (a)-(e) are sub cases of Case 1. So nothing survives out of Case 4 at order $\ep^5$.

For quadratic equations, when $\xi_1=\xi_2=\xi_3=\xi_4=\zeta=0$ in equations $\CQ^+$, the
$\ep^5$ order necessary conditions for the integrability of
the resulting equation read:

* Case Q$_{\alpha}$: $\alpha_{1}=\alpha_{2}=\beta_{2}=\gamma_{1}=0$, $\beta_{1}=\gamma_{2}\not=0$, $a_{2}=2a_{1}$;

* Case Q$_{\beta}$: $\alpha_{1}=\alpha_{2}=\beta_{2}=\gamma_{2}=0$, $\beta_{1}=\gamma_{1}\not=0$, $a_{2}=2a_{1}$;

* Case Q$_{\gamma}$: $\beta_{1}=\alpha_{2}=\beta_{2}=\gamma_{1}=0$, $\alpha_{1}=\gamma_{2}\not=0$, $a_{2}=2a_{1}$;

* Case Q$_{\delta}$: $\beta_{1}=\alpha_{2}=\beta_{2}=\gamma_{2}=0$, $\alpha_{1}=\gamma_{1}\not=0$, $a_{2}=2a_{1}$;

* Case Q$_{\eta}$: $\alpha_{1}=\alpha_{2}=\beta_{2}=\gamma_{1}=0$, $\beta_{1}=\gamma_{2}\not=0$, $a_{1}=2a_{2}$;

* Case Q$_{\theta}$: $\alpha_{1}=\alpha_{2}=\beta_{2}=\gamma_{2}=0$, $\beta_{1}=\gamma_{1}\not=0$, $a_{1}=2a_{2}$;

* Case Q$_{\kappa}$: $\beta_{1}=\alpha_{2}=\beta_{2}=\gamma_{1}=0$, $\alpha_{1}=\gamma_{2}\not=0$, $a_{1}=2a_{2}$;

* Case Q$_{\lambda}$: $\beta_{1}=\alpha_{2}=\beta_{2}=\gamma_{2}=0$, $\alpha_{1}=\gamma_{1}\not=0$, $a_{1}=2a_{2}$.

The Cases Q$_{\alpha}$-Q$_{\delta}$ are sub cases both of the Case Q1 and Case (a); the Cases Q$_{\eta}$-Q$_{\lambda}$ are sub cases both of the Case Q1 and Case (b).

§.§.§ Canonical forms for $\varepsilon^5$ asymptotically integrable cases.

We will now use the Möbius transformation to reduce the equation to normal form, i.e. to eliminate the maximum number of free parameters appearing in the nonlinear difference equation and reduce the coefficients of the linear part in $v_{n,m}$ and $v_{n+1,m+1}$ to 1.

In the Case (a) of Proposition 4, performing the Möbius transformation
β=0,    γ=-γ_1δ/2,    α=a_1δ,   δ≠0,
we obtain the canonical form:

Case (a$^{\prime}$): ($\CN=2$)

where $(\tau_{1}, \tau_{2})\doteq\left(\alpha_{1}-\frac{3\gamma_{1}}{2}, \beta_{1}-\frac{3\gamma_{1}}{2}\right)\not=(0, 0)$. Performing a further rescaling  on (<ref>), we can fix, in all generality, the coefficients to either  $\tau_{1}=0$ and $\tau_{2}=1$ or $\tau_{1}=1$ with $\tau_{2}$ arbitrary and we obtain  the following two canonical forms respectively

representing the two non overlapping subclasses of Case (a) defined respectively by the additional conditions $\alpha_{1}=\frac{3\gamma_{1}}{2}$ and $\alpha_{1}\not=\frac{3\gamma_{1}}{2}$. As under a restricted Möbious transformation $\tau_{2}$ is invariant, we see that two canonical forms (<ref>), specified by two invariants $\tau_{2a}$ and $\tau_{2b}$, form two disconnected components of the same conjugacy subclass unless $\tau_{2a}=\tau_{2b}$.

In the Case (b) of Proposition 4, performing the Möbius transformation
β=0,    γ=-γ_2δ/2,    α=a_2δ,   δ≠0,
we obtain the canonical form:

Case (b$^{\prime}$): ($\CN=2$)

where $(\tau_{1}, \tau_{2})\doteq\left(\alpha_{1}-\frac{3\gamma_{2}}{2}, \beta_{1}-\frac{3\gamma_{2}}{2}\right)\not=(0, 0)$. Performing a further rescaling on (<ref>) we can fix, in all generality, the parameters either to $\tau_{1}=0$ and $\tau_{2}=1$ or to $\tau_{1}=1$ with $\tau_{2}$ arbitrary and  we obtain respectively the two canonical forms

representing the two non overlapping subclasses of Case (b) defined respectively by the additional conditions $\alpha_{1}=\frac{3\gamma_{2}}{2}$ and $\alpha_{1}\not=\frac{3\gamma_{2}}{2}$. As  $\tau_{2}$ is invariant under a restricted Möbious transformation, we see that two canonical forms (<ref>), specified by two invariants $\tau_{2a}$ and $\tau_{2b}$, form two disconnected components of the same conjugacy subclass unless $\tau_{2a}=\tau_{2b}$.

In the Cases (c) and (d) of Proposition 4, performing the  Möbius transformation
α=2a_1δ/(a_1+a_2)√(|ρ|),   β=0,   γ=-γ_1δ/(a_1+a_2)√(|ρ|),   δ≠0,
we obtain the canonical forms:

Case (c$^{\prime}$): ($\CN=2$)

Case (d$^{\prime}$): ($\CN=2$)

where $\epsilon\doteq a_{2}/a_{1}\not=0,\pm 1$ and $\zeta^{\prime}\doteq 8s\left|\frac{\pi^2}{\rho^3}\right|^{1/2}/\left(1+\epsilon\right)^2$, $\pi\doteq\left[\zeta-2\frac{\gamma_{1}} {a_{1}}\xi_{2}+\frac{\left(a_{1}+a_{2}\right)\gamma_{1}^3} {2a_{1}^3}\right]/\left(a_{1}+a_{2}\right)$ and $s\doteq\pm 1$. As under a restricted Möbius transformation $\rho\rightarrow\rho\left(\alpha/\delta\right)^2$ and $\pi\rightarrow\pi\left(\alpha/\delta\right)^3$, we see that the absolute value of $\zeta^{\prime}$ and $\operatorname{sgn}\left(\rho\right)$ are invariant under such a transformation.  With another rescaling we can always fix $\zeta^{\prime}\geq 0$ and the two canonical forms, specified by the two set of invariants $\left(\epsilon_{a}, \operatorname{sgn}\left(\rho_{a}\right), \zeta^{\prime}_{a}\right)$ and $\left(\epsilon_{b}, \operatorname{sgn}\left(\rho_{b}\right), \zeta^{\prime}_{b}\right)$, form two disconnected components of the conjugacy class unless the two sets are  the same.

In the Case (e) of Proposition 4, performing the Möbius transformation
β=0,    γ=-(γ_1+γ_2)α/2(a_1+a_2),    δ=(a_2γ_1-a_1γ_2)α/a_1(a_1+a_2),   α≠0,
we obtain the canonical form:

Case (e$^{\prime}$): ($\CN=1$)

where $\epsilon\doteq a_{2}/a_{1}\not=0,\pm 1,2,1/2$. As $\epsilon$ is invariant under a restricted Möbius transformation, we see that two canonical forms, specified by the two invariants $\epsilon_{a}$ and $\epsilon_{b}$, form two disconnected components of the conjugacy class unless $\epsilon_{a}=\epsilon_{b}$.

§.§.§ Comparison with the ABS list.

As our allowed transformations are sub cases of the full Möbius transformations allowed in the ABS approach [2], any conjugacy class of ours is either completely contained into one of the ABS classification or is totally disjointed from them. Considering that no one out of the  canonical forms (a$^{\prime}$)-(e$^{\prime}$) possesses the invariance (up to an overall sign) under the transformation $v_{n,m}\leftrightarrow v_{n+1,m}$, $v_{n,m+1}\leftrightarrow v_{n+1,m+1}$, we can conclude that no intersection can exist between our classes and those generated by the $ABS$ list. Even more, no equation in our list is of Klein-type or, that is the same [11], a sub case of the $Q_{V}$ equation.

We can  enlarge our class of transformations by including also an exchange $n\leftrightarrow m$ between the two independent variables. The subclass (<ref>) can be discarded because under this exchange we would get it from subclass (<ref>) with $\tau_{2}=0$; similarly the subclass (<ref>) can be discarded because under this exchange we would get it from subclass (<ref>) with $\tau_{2}=0$; finally the subclasses (<ref>-<ref>) are invariant under this transformation.

Let us include also  the inversion $n\rightarrow -n$. Setting $\tilde v_{n,m}\doteq v_{-n,m}$, we have that, if $v_{n,m}$ satisfies (<ref>), then $\tilde v_{n,m}$ satisfies (<ref>); if $v_{n,m}$ satisfies (<ref>) with parameters $\epsilon$ and $\zeta^{\prime}$, then $\tilde v_{n,m}\doteq \operatorname{sgn}\left(\epsilon\right)v_{-n,m}$ satisfies (<ref>) with parameters $1/\epsilon$ and $\zeta^{\prime}/\left|\epsilon\right|$ and similarly for Eq. (<ref>); if $v_{n,m}$ satisfies (<ref>) with parameter $\epsilon$, then $\tilde v_{n,m}\doteq -v_{-n,m}/\epsilon$ satisfies (<ref>) with parameter $1/\epsilon$ (this implies that, if $v_{n,m}$ satisfies one of the four canonical forms (<ref>), (<ref>-<ref>), then also $\tilde v_{n,m}\doteq v_{-n,-m}$ does). As a consequence, under this enlarged class of transformations,  (<ref>) can be discarded and in the case of (<ref>-<ref>) we can limit the p!
arameter $\epsilon$ to the range $-1<\epsilon<1$, $\epsilon\not=0$ as the equation with parameters $1/\epsilon$ and $\zeta^{\prime}$ can be obtained from the corresponding with parameters $\epsilon$ and $\zeta^{\prime}\left|\epsilon\right|$.

Remarque: The Cases (c') and (d'), when $\pi=0$, i.e. $\zeta^{\prime}=0$, reduce to the integrable cases analyzed in Levi-Yamilov [11] and Ramani-Grammaticos [14].

§.§ Classification at order $\ep^6$.

Now we perform a multiscale reduction at order $\ep^6$ on the four canonical forms (<ref>), (<ref>-<ref>) and  we find that all the so far obtained equations satisfy the $A_{4}$-integrability conditions (<ref>). Hence we can state the following proposition

Up to a restricted Möbius transformations $\tilde v_{n,m}\doteq v_{n,m}/\left(\alpha v_{n,m}+\beta\right)$, exchanges $n\leftrightarrow m$ and inversions $n\rightarrow -n$, all the $A_{4}$-asymptotically integrable cases in the class ${\mathcal Q}^{+}$ are given by

+τv_n,mv_n+1,mv_n,m+1v_n+1,m+1=0,   -1<ϵ<1,   ϵ≠0,   δ≐±1,   τ≥0;

+τv_n,mv_n+1,mv_n,m+1v_n+1,m+1=0,   -1<ϵ<1,   ϵ≠0,   δ≐±1,   τ≥0;

+(1-1/ϵ^2)v_n,mv_n+1,mv_n,m+1v_n+1,m+1=0,   -1<ϵ<1,   ϵ≠0, 1/2.

If, when $\tau=0$  in (<ref>), we apply the (not allowed) transformation $v_{n,m}\doteq\sqrt{3}w_{n,m}-1$, we obtain
which in the direction $n$ satisfies two necessary integrability conditions given in [11] but does't admit any three-points generalized symmetries either autonomous or not, while in the direction $m$ the integrability conditions given in [11] are not satisfied;

If, when $\tau=1$ in (<ref>), we apply the (not allowed) transformation $v_{n,m}\doteq 2^{1/3}w_{n,m}-1$, we obtain
an integrable equation considered in [12], which possesses a $3\times 3$ Lax pair and which is a degeneration of the discrete integrable Tzitzeica equation proposed by Adler in [1].

Finally, if we choose $\tau\not=0$, $1$  in (<ref>), we can apply the (not allowed) transformation $v_{n,m}\doteq \frac{1-\tau}{\tau}w_{n,m}-1$ and we obtain
where $\chi\doteq\frac{\left(\tau-3\right)\tau^2}{\left(1-\tau\right)^3}$. This system is the sum of (<ref>), (<ref>) and an arbitrary constant and doesn't satisfy the integrability conditions given in [11] for three-points generalized symmetries either autonomous or not, either in the direction $n$ or $m$.

If in (<ref>), (<ref>) we apply respectively the (not allowed) transformations $w_{n,m}\doteq\delta \operatorname{sgn}\left(\epsilon\right)/v_{n,m}$ and $\tilde w_{n,m}\doteq\delta/v_{n,m}$, we obtain

Eqs.(<ref>, <ref>) are just an almost trivial looking  modification of the two integrable systems discussed in [14], which are recovered when $\tau=0$. In [14] it  was shown that, when $\tau=0$, (<ref>, <ref>) are mapped through a Möbious transformation respectively to the Hirota discrete sine-Gordon equation and to its potential form. After we replace $\epsilon\rightarrow 1/\epsilon$ in (<ref>) and $\delta\rightarrow s\delta$, with $s\doteq \operatorname{sgn}\left(\epsilon\right)$,  in (<ref>) the precise form of the potentiation induced between them is
Eqs.(<ref>, <ref>), if and only if $\tau=0$, satisfy the integrability conditions given in [11] for three-points generalized symmetries either autonomous or not which, in the $n$ direction, are given respectively by
w_n,m,t   =   (δw_n,m^2-ϵ)(δϵw_n,m^2-1)(w_n+1,m-w_n-1,m)/(1+δw_n,mw_n+1,m)(1+δw_n,mw_n-1,m),

w̃_n,m,t̃   =   Y(δw̃_n,m^2-1)(w̃_n+1,m-w̃_n-1,m)/δw̃_n+1,mw̃_n-1,m-1+[(-1)^nκ+(-1)^mθ](δw̃_n,m^2-1),
where $t$ and $\tilde t$ are two group parameters, and, in the $m$ direction, by expressions obtained changing $w_{n+1,m}\rightarrow w_{n,m+1}$ and $w_{n-1,m}\rightarrow w_{n,m-1}$. When $\tau=0$ (<ref>) admits a two parameters non autonomous point symmetry. Eqs. (<ref>, <ref>) are invariant under the transformation $w_{n,m}\doteq -v_{n,m}$; Eq. (<ref>) is covariant under the inversion $w_{n,m}\doteq 1/v_{n,m}$ as $\epsilon$ is changed into $1/\epsilon$, while (<ref>) is invariant.  Eqs. (<ref>, <ref>), under the non autonomous transformation $w_{n,m}\doteq \left(-1\right)^{n+m}v_{n,m}$, are covariant as in the first case $\epsilon$ is changed into $-\epsilon$ and $\delta$ into $-\delta$ while in the second one $\epsilon$ is changed into $-\epsilon$, implying that we can limit ourselves to the range $0<\epsilon<1$. Eq. (<ref>) is invariant under the non autonomous transformation $\tilde w_{n,m}\doteq [v_{n,m}]^{\left(-1\right)^{n+m}}$ when $\delta=1$ !
and covariant when $\delta=-1$ as $\epsilon$ is changed into $\delta\epsilon$. Finally (<ref>, <ref>) are covariant under the transformation $w_{n,m}\doteq\ri v_{n,m}$ as $\delta$ is changed into $-\delta$, this implying that we can always take $\delta=1$ even if in general, allowing such a transformation, the solution will be no more real.

If  we apply the (not allowed) transformation $v_{n,m}\doteq\frac{|\rho|^{1/2}w_{0,0}+1}{|\rho|^{1/2}w_{0,0}-1/\epsilon}$, with $\rho\doteq\frac{-1+2\epsilon}{\epsilon\left(\epsilon-2\right)}\not=0$, to (<ref>) we obtain

where $\delta\doteq \operatorname{sgn}\left(\rho\right)=\operatorname{sgn}\left(1/\epsilon-2\right)$, which, for $\delta=-1$, is a real discrete Tzitzeica equation with coefficient $c=1/\left(\epsilon|\rho|^{3/2}\right)$ and, for $\delta=1$, through the (not allowed) transformation $w_{n,m}\rightarrow\ri w_{n,m}$ becomes a complex Tzitzeica equation with coefficient $c=\ri/\left(\epsilon|\rho|^{3/2}\right)$. We remember that the discrete Tzitzeica equation is integrable and it admits a $3\times 3$ Lax pair [1].

We note that (<ref>), beside not being sub cases of the $Q_{V}$ equation, except for (<ref>, <ref>) with $\tau=0$, where a five-points generalized symmetry depending on the points $\left(n+1,m\right)$, $\left(n,m+1\right)$, $\left(n,m\right)$, $\left(n-1,m\right)$ and $\left(n,m-1\right)$ exists, is also not included into the Garifullin-Yamilov class [7].

§ CONCLUDING REMARKS

In this paper we have considered the application of a multiple scale expansion  to the class of dispersive multilinear partial difference equation on the square lattice, $\CQ$. A great effort has been directed to extend the expansion up to order $\ep^6$ so as to be able to check the order $\ep^5$ results. The integrability conditions we obtain, when we require that the multiple scale expansion of the discrete class of equations is equivalent to the equations of the NLSE hierarchy, reduce the $13-$parameters initial class to four equations depending on few parameters. The $A_3$ integrable equations are invariant under the $A_4$ integrability conditions, indicating that we might have already filtered out all the nonintegrable cases and that the obtained equations might be integrable.

Two open problems seem of great importance now:

* The consideration of the second class of dispersive multilinear partial difference equations on the square lattice, $\CQ=\CQ^-$ is  will provide by sure new classes of integrable equations, as in this case the lowest order integrability conditions appear already at order $\ep^2$ and will not be an equation of the NLSE type but more likely a coupled wave equations.

* We may assume that (<ref>) are all integrable but in this article we are not able to prove it. We have to apply different techniques. In particular what remains to be analyzed are the four sub cases (<ref>), (<ref>) and (<ref>) when $\tau>0$. We are now working for proving that these systems have generalized symmetries, i.e. there exist some flows in the group parameter $\lambda$ commuting with our equations

$$v_{n,m,\lambda}=g(v_{n+k,m}, v_{n+k-1,m}, \cdots, v_{n-k,m}, v_{n,m+\ell}, v_{n,m+\ell-1}, \cdots, v_{n,m-\ell}).$$
It is easy to prove that there are no symmetries with $k=\ell=1$. Thus it seems important to consider  the case when $k=\ell\ge 2$.

Work is in progress in both open problems. In particular in [15] one has proved the integrability of (<ref>) by constructing two generalized symmetries defined on five points, one with $k=2$ and $\ell=0$ and one with $k=0$ and $\ell=2$.

§ ACKNOWLEDGMENTS

LD and SC have been partly supported by the Italian Ministry of Education and Research, PRIN
“Nonlinear waves: integrable fine dimensional reductions and discretizations" from 2007
to 2009 and PRIN “Continuous and discrete nonlinear integrable evolutions: from water
waves to symplectic maps" from 2010. RHH thanks the INFN, Sezione Roma Tre and the UPM for their support during his visits to Rome. We thank Matteo Petrera for many enlightening discussion in the first stage of this paper.

§ APPENDIX

Here we present explicitly the $48$ conditions for $\varepsilon^6$ $S$-asymptotic integrability ($A_{4}$-integrability) involving the real ($S_{j}$) and imaginary parts ($T_{j}$) of the coefficients $\eta_{j}$, $j=1$,…, $34$ of the differential polynomial (<ref>). The expressions of the coefficients $\kappa_{m}$ , $m=1$,…, $77$ of the differential polynomial (<ref>) as functions of the $\eta_{j}$, $j=1$,…, $34$ are complicated, so we will omit them. These $48$ conditions are, as far as we know, presented here for the first time.

\begin{align}\label{Phoenix}
T_{9}&=T_{10}-\frac{I_{1}T_{32}}{2\rho_{2}}-\frac{I_{6}T_{33}}{4\rho_{1}},\qquad S_{11}=\frac{S_{22}}{2}-\left[\frac{R_{12}}{2}+\left(a-b\right)\frac{I_{8}}{\rho_{2}}\right]\frac{T_{32}}{2\rho_{2}},\qquad T_{11}=T_{22}+\frac{I_{8}T_{27}-\left(I_{6}+I_{12}\right)T_{32}}{2\rho_{2}},\nonumber\\
S_{12}&=-\frac{I_{8}S_{27}}{\rho_{2}}+\left(\frac{a}{2}-b\right)\frac{I_{8}T_{32}}{\rho_{2}^2},\qquad T_{12}=-\frac{I_{8}\left(T_{27}-T_{33}\right)}{\rho_{2}},\nonumber\\
S_{14}&=0,\qquad T_{14}=-\frac{I_{8}T_{32}}{2\rho_{2}},\nonumber\\
S_{16}&=-\frac{I_{6}S_{27}+aT_{22}}{2\rho_{2}}+\left(a-\frac{3b}{2}\right)\frac{I_{6}T_{32}}{2\rho_{2}^2}+\frac{R_{12}T_{33}}{2\rho_{2}},\qquad T_{16}=\frac{aS_{22}-I_{6}T_{27}}{2\rho_{2}}+\left(\frac{aR_{12}}{2\rho_{2}}-I_{3}\right)\frac{T_{32}}{2\rho_{2}}+\left(I_{6}+I_{12}\right)\frac{T_{33}}{2\rho_{2}},\nonumber\\
S_{17}&=S_{15}+\frac{I_{12}S_{27}}{2\rho_{2}}-\left(a-b\right)\frac{I_{8}T_{27}}{2\rho_{2}^2}+\left[\left(2a+3b\right)I_{6}+bI_{12}\right]\frac{T_{32}}{4\rho_{2}^2}+\left(a-\frac{5b}{4}\right)\frac{I_{8}T_{33}}{\rho_{2}^2},\qquad T_{17}=-\frac{aS_{22}}{2\rho_{2}}+\nonumber\\
T_{19}=T_{20},\qquad S_{21}=S_{22},\qquad T_{21}=T_{22},\nonumber\\
S_{26}&=-\frac{I_{8}S_{27}}{\rho_{2}}-\left(R_{12}+\frac{aI_{8}}{\rho_{2}}\right)\frac{T_{32}}{2\rho_{2}},\qquad T_{26}=T_{22}-\frac{I_{6}T_{32}-I_{8}T_{33}}{\rho_{2}},\qquad
S_{30}=0,\qquad T_{30}=T_{32},\nonumber\\
S_{31}&=0,\qquad T_{31}=T_{32},\qquad S_{32}=0,\qquad S_{33}=-\frac{aT_{32}}{2\rho_{2}},\qquad S_{34}=S_{27}+\frac{bT_{32}}{2\rho_{2}},\qquad T_{34}=0,\nonumber\\
&\quad{}+2\left[2\rho_{2}\left(a-b\right)R_{12}+2\rho_{2}^2\left(I_{2}-I_{3}-2I_{5}\right)+a\left(b-2a\right)I_{8}\right]T_{33}=0,\qquad I_{6}T_{32}-I_{8}T_{33}=0.\nonumber
\end{align}

[1]V.E. Adler, On a discrete analog of the Tzitzeica equation, (arXiv:1103.5139);
[2]
Adler V.E., Bobenko A.I. and Suris Yu.B., Classification of integrable equations on quad-graphs. The consistency approach, Comm. Math. Phys. 233/3, 513–543 (2003);

[3]
A. Degasperis, Multiscale expansion and integrability of dispersive wave equations, lectures given at the Euro Summer School "What is integrability?", Isaac Newton Institute, Cambridge, U. K., 13-24 August 2001, in Integrability edited by A. V. Mikhailov, Lecture Notes in Physics 767, 215-244, Springer, Berlin-Heidelberg (2009);

[4]
A. Degasperis, S. V. Manakov and P.M. Santini, Multiple-scale perturbation beyond the nonlinear Schrödinger equation. I, Phys. D 100, 187–211 (1997);

[5] A. Degasperis and M.  Procesi, , "Asymptotic integrability", in A. Degasperis and G. Gaeta,  Symmetry and Perturbation Theory (Rome, 1998), River Edge, NJ: World Scientific  (1999),  23?37;

[6]R.N. Garifullin, E.V. Gudkova, I.T. Habibullin, Method for searching higher symmetries for quad graph equations, J. Phys. A: Math. Theor. 44 (2011) 325202 (ArXiv:1104.0493);

[7]R.N. Garifullin, R.I. Yamilov, Generalized symmetry classification of discrete equations of a class depending on twelve parameters, (arXiv:1203.4369);

[8]
R. Hernandez Heredero, D. Levi, C. Scimiterna, A discrete linearizability test based on multiscale analysis, Jour. Phys. A: Math. and Theor. 43 (2010), 502002;

[9]Y. Hiraoka, Y. Kodama, Normal forms and solitons, in Integrability, A.V. Mikhailov editor, Lecture Notes in Physics Volume 767, Springer, Berlin 2009,  175-214;

[10]D. Levi and R.I. Yamilov, On a nonlinear integrable difference equation on the square, Ufa Math. J. 1:2 (2009) 101-105. (arXiv:0902.2126v2);

[11]D. Levi and R.I. Yamilov, Generalized symmetry integrability test for discrete equations on the square lattice, J. Phys. A: Math. Theor. 44 (2011) 145207;

[12]A.V. Mikhailov and P. Xenitidis,  Second order integrability conditions for difference equations. An integrable equation, arXiv:1305.4347;

[13]
Santini P.M., The multiscale expansions of difference equations in the small lattice spacing regime, and a vicinity and integrability test: I, Jour. Phys. A: Math. and Theor. 43 (2010), 045209;

[14]
C. Scimiterna, B. Grammaticos, A. Ramani, On two integrable lattice equations and their interpretation, Jour. Phys. A: Math. and Theor., 44, no. 3 (2011), 032002.

[15]
C. Scimiterna, M. Hay and D. Levi, On the integrability of the lattice equation $w_{n,m}w_{n+1,m}+w_{n+1,m}w_{n,m+1}+w_{n,m+1}w_{n+1,m+1}=1$, work in progress

[16] C. Viallet, Integrable lattice maps: $Q_V$, a rational version of $Q_4$, Glasgow Math. J., 51A (2009)  157-163, (arXiv:0802.0294v1);

\begin{multline}
\text{I2.\ }2 a_2(
u_{n,m+1}+ u_{n+1,m})+4 a_2d\, u
u_{n+1,m+1}+2a_2d\, u_{n,m+1}
\\
{}+3 a_2d\,(
u_{n+1,m+1}+u_{n+1,m} u_{n+1,m+1}
+u u_{n,m+1}+u
\\
_2+\tau _4\right)u u_{n+1,m+1} (
_2-\tau _4\right) u_{n,m+1}
u_{n+1,m} (u+u_{n+1,m+1})
\end{multline}

\begin{multline}
\text{I3.\ }
\frac{1}{2} a_2
u u_{n+1,m+1}+2a_2d\, u_{n,m+1}
\\
{}+\frac{3}{2} a_2d(
u_{n+1,m} u_{n+1,m+1}+
u u_{n,m+1}+u
\\
_2+\tau _4\right) u
_2+\tau _4\right) u
_2-\tau _4\right) u_{n,m+1}
u_{n+1,m} u_{n+1,m+1}
\\
{}+\left(\tau _2-\tau
_4\right) u u_{n,m+1}
\end{multline}

* Case 1-4.
$\left\{\alpha_2=\beta_2=0,\ \alpha _1= \beta _1 =
\frac{1}{2}
\left(\gamma_1+d_2\right),\ a_2d_1 = a_1d_2,\ \tau _1= \tau_2,\ \tau _3=\tau_4\right\}$

* Case 1-2. $\left\{\alpha_2=\beta_2=0,\ \alpha _1= \beta
_1,\ d_1= 2 d_2,\ a_1= 2 a_2,\ \tau_1=\tau_2,\ \tau _3= \tau_4\right\}$

* Case 1-3. $\left\{\alpha_2=\beta_2=0,\ \alpha _1= \beta
_1,\ 2d_1= d_2,\ 2a_1= a_2,\ \tau _1=\tau_2,\ \tau _3= \tau_4\right\}$

* Case 2-4. $\left\{\alpha _2= \beta
_2,\ \alpha _1= \beta
_1= \frac32
d_2,\ d_1= 2
d_2,\ a_1= 2 a_2,\ \tau _1= \tau
_2-\frac{d_1 \alpha
_2}{a_1},\tau _3=
\frac{d_1 \alpha
_2}{a_1}+\tau _4\right\}$

* Case 3-4. $\left\{\alpha _2=
-\beta _2,\ \alpha _1=
\beta _1=
\frac34 d_2,\ 2d_1=
d_2,\ 2a_1=
a_2,\ \tau _1=
\tau _2-\frac{d_1 \alpha
_2}{a_1},\ \tau _3= \tau
_4-\frac{d_1 \alpha

* Case 1-5.
$\left\{\alpha_2=\beta_2=0,\ \alpha _1=
\frac{\left(a_1+a_2\right)
\left(a_1^2 d_2-a_2^2
d_1\right)}{2 a_1
\left(a_1-a_2\right)
a_2},\ \beta
_1= \frac{a_1^2 d_2+a_2^2
d_1}{2 a_1 a_2},\
\tau _1= \tau _2,\ \tau
_3= \tau _4\right\}$

* Case 1-6. $\left\{\alpha_2=\beta_2=0,\ \alpha _1=
\frac{a_1^2 d_2+a_2^2
d_1}{2 a_1 a_2},\ \beta _1=
\frac{\left(a_1+a_2\right)
\left(a_1^2 d_2-a_2^2
d_1\right)}{2 a_1
\left(a_1-a_2\right)
a_2},\ \tau
_1= \tau _2,\ \tau _3=
\tau _4\right\}$

* Case 2-5.
$\left\{\alpha_2=\beta_2=0,\ \alpha _1=\beta_1= \frac32 d_2,\ d_1= 2
d_2,\ a_1=2 a_2,\ \tau _1=\tau_2,\ \tau _3= \tau

* Case 2-6.
$\left\{\alpha_2=\beta_2=0,\ \alpha _1=\beta_1= \frac32 d_2,\ d_1= 2
d_2,\ a_1=2 a_2,\ \tau _1=\tau_2,\ \tau _3= \tau

* Case 3-5.
$\left\{\alpha_2=\beta_2=0,\ \alpha _1=\beta _1= \frac34
d_2,\ 2d_1=d_2,\ 2a_1=a_2,\ \tau
_1= \tau _2,\ \tau _3=
\tau _4\right\}$

* Case 3-6.
$\left\{\alpha_2=\beta_2=0,\ \alpha _1=\beta _1= \frac34
d_2,\ 2d_1=d_2,\ 2a_1=a_2,\ \tau
_1= \tau _2,\ \tau _3=
\tau _4\right\}$

* Case 4-5. $\left\{\alpha _2=
\frac{a_2-a_1}{a_1+a_2}\beta _2,\ \alpha _1=
\beta _1=
\frac{a_1+a_2}{2 a_2}d_2,\ a_2d_1=a_1
d_2,\ \tau _1= \tau
_2-\frac{d_2 \alpha
_2}{a_2},\ \tau _3=
\frac{d_2 \beta
_2}{a_2}+\tau _4\right\}$

* Case 4-6. $\left\{\alpha _2=
\frac{a_1+a_2
}{a_2-a_1}\beta _2,\ \alpha _1=\beta _1=
\frac{a_1+a_2}{2 a_2}d_2,\ a_2d_1= a_1d_2,\ \tau _1= \tau
_2-\frac{d_2 \alpha
_2}{a_2},\ \tau _3=
\frac{d_2 \beta
_2}{a_2}+\tau _4\right\}$

* Case 1-2-5. $\left\{\alpha_2=\beta_2=0,\ \alpha _1=
\beta _1= \frac{3
d_2}{4},\ 2d_1=d_2,\ 2a_1=a_2,\ \tau
_1= \tau _2,\ \tau _3=
\tau _4\right\}$

* Case 1-3-5. $\left\{\alpha_2=\beta_2=0,\ \alpha _1=
\beta _1= \frac{3
d_2}{2},\ d_1=2d_2,\ a_1=2a_2,\ \tau
_1= \tau _2,\ \tau _3=
\tau _4\right\}$

* Case 4:

\left\{ \begin{array}{l}
{\displaystyle{\frac{a_2}{a_1} }}= -3.309 , \\
\alpha_1 = 0.081 \delta_2 - 0.887 \delta_1, \\
\beta_1 = -1.655 \delta_1 - 0.151 \delta_2, \\
{\displaystyle{ \frac{\beta_2}{\alpha_2}}} = 0.536, \\
{\displaystyle{  \frac{\tau_1 - \tau_2}{\alpha_2}}}=
{\displaystyle{ \frac{ 0.232(  \delta_2 - \delta_1) }{a_1}}}, \\
{\displaystyle{  \frac{\tau_3 - \tau_4}{\alpha_2}}}=-
{\displaystyle{ \frac{0.588 \delta_1 + 0.339
\delta_2}{a_1}}}.
\end{array} \right.

* Case 5:

\left\{ \begin{array}{l}
{\displaystyle{\frac{a_2}{a_1} }}= 0.302 , \\
\alpha_1 = 0.081 \delta_1 - 0.887 \delta_2, \\
\beta_1 = -1.655 \delta_2 - 0.151 \delta_1, \\
{\displaystyle{ \frac{\beta_2}{\alpha_2}}} = -0.536, \\
{\displaystyle{  \frac{\tau_1 - \tau_2}{\alpha_2}}}=
{\displaystyle{ \frac{ 0.768 (  \delta_2 - \delta_1) }{a_1}}}, \\
{\displaystyle{  \frac{\tau_3 - \tau_4}{\alpha_2}}}=-
{\displaystyle{ \frac{1.124 \delta_1 + 1.984
\delta_2}{a_1}}}.
\end{array} \right.

* Case 6:

\left\{ \begin{array}{l}
{\displaystyle{\frac{a_2}{a_1} }}= -2.708  , \\
\alpha_1 = -0.185 \delta_2 - 1.354 \delta_1, \\
\beta_1 = 0.085 \delta_2 - 0.624 \delta_1, \\
{\displaystyle{ \frac{\beta_2}{\alpha_2}}} = 2.171, \\
{\displaystyle{  \frac{\tau_1 - \tau_2}{\alpha_2}}}=
{\displaystyle{ \frac{ 1.440 \delta_1 +0.901 \delta_2 }{a_1}}}, \\
{\displaystyle{  \frac{\tau_3 - \tau_4}{\alpha_2}}}=
{\displaystyle{ \frac{0.585 ( \delta_1 -
\delta_2)}{a_1}}}.
\end{array} \right.

* Case 7:

\left\{ \begin{array}{l}
{\displaystyle{\frac{a_2}{a_1} }}= -0.369  , \\
\alpha_1 = -0.185 \delta_1 - 1.354 \delta_2, \\
\beta_1 = 0.085 \delta_1 - 0.624 \delta_2, \\
{\displaystyle{ \frac{\beta_2}{\alpha_2}}} = -2.171, \\
{\displaystyle{  \frac{\tau_1 - \tau_2}{\alpha_2}}}=
-{\displaystyle{ \frac{ 2.440 \delta_1 + 3.901\delta_2 }{a_1}}}, \\
{\displaystyle{  \frac{\tau_3 - \tau_4}{\alpha_2}}}=
{\displaystyle{ \frac{1.585 ( \delta_2 -
\delta_1)}{a_1}}}.
\end{array} \right.