On the Classification of Darboux Integrable Chains

We study differential-difference equation of the form $t_{x}(n+1)=f(t(n),t(n+1),t_x(n))$ with unknown $t=t(n,x)$ depending on $x$, $n$. The equation is called Darboux integrable, if there exist functions $F$ (called an $x$-integral) and $I$ (called an $n$-integral), both of a finite number of variables $x$, $t(n)$, $t(n\pm 1)$, $t(n\pm 2)$, $...$, $t_x(n)$, $t_{xx}(n)$, $...$, such that $D_xF=0$ and $DI=I$, where $D_x$ is the operator of total differentiation with respect to $x$, and $D$ is the shift operator: $Dp(n)=p(n+1)$. The Darboux integrability property is reformulated in terms of characteristic Lie algebras that gives an effective tool for classification of integrable equations. The complete list of equations of the form above admitting nontrivial $x$-integrals is given in the case when the function $f$ is of the special form $f(x,y,z)=z+d(x,y)$.