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\newarticle{The quad graph equation with a nonstandard generalized symmetry structure.}{R.N.~Garifullin, A.V.~Mikhailov and R.I.~Yamilov.}{
Institute of Mathematics, Ufa, Russia\\ 
Department of Applied Mathematics, University of Leeds, United Kingdom\\e-mail: rustem@matem.anrb.ru} 


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The equation \begin{equation} u_{n+1,m+1}(u_{n,m}-u_{n,m+1})-u_{n+1,m}(u_{n,m}+u_{n,m+1})+1=0\label{gar_tzit}\end{equation} is found in article \cite{GY12:Garufullin}. In those artcile is shown that eq.(\ref{gar_tzit}) have two generalized symmetry in different directions:
\begin{equation}\label{gar_s2_tz}\frac{d}{dt_1}u_{n,m}=h_{n,m}h_{n-1,m}(a_nu_{n+2,m} - a_{n-1}u_{n-2,m}),\end{equation} 
\begin{equation}{\frac{d}{dt_2}u_{n,m}=(-1)^n\frac{u_{n,m+1}u_{n,m-1}+u_{n,m}^2}{u_{n,m+1}+u_{n,m-1}},\label{gar_s1_tz}}\end{equation}where
$h_{n,m}=1-2u_{n+1,m}u_{n,m},\quad a_{n+2}=a_{n}.$ One can see that $n$ is an outer parameter in eq. (\ref{gar_s1_tz}), and this equation is really a known 1+1-dimensional autonomous equation of the Volterra type. In the case of eq. (\ref{gar_s2_tz}), we have the essentially non-autonomous Itoh-Narita-Bogoyavlensky equation with two-periodic coefficient $a_n$. 

We show that eq. (\ref{gar_s1_tz}) can be rewritte as Gerdjikov-Ivanov-Tsuchida system \cite{T02:Garifullin} for odd and even $u_{n,m}$. We find Lax pairs for equations (\ref{gar_tzit},\ref{gar_s2_tz},\ref{gar_s1_tz}) in the form: $$\Psi_{n+2,m}=N_{n,m}\Psi_{n,m},\quad \Psi_{n,m+1}=M_{n,m}\Psi_{n,m} $$  $$\frac{d}{dt_1}\Psi_{n,m}=A_{n,m}\Psi_{n,m},\quad \frac{d}{dt_2}\Psi_{n,m}=B_{n,m}\Psi_{n,m},$$ where $\Psi_{n,m}$ -- vector function, $A_{n,m},B_{n,m},N_{n,m},M_{n,m}$ -- $2\times 2$ matrtixes. 

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\begin{thebibliography}{99}
\bibitem{GY12:Garufullin} R.N.~Garifullin and R.I.~Yamilov Generalized symmetry classification of discrete equations of a class depending on twelve parameters 2012 J. Phys. A: Math. Theor. 45 345205.

\bibitem{T02:Garifullin} T. Tsuchida Integrable discretizations of derivative nonlinear Schrodinger equations, 2002 J. Phys. A: Math. Gen. 35 7827.
\end{thebibliography}
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