  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 $f(x,y,z)=z+d(x,y)$.
