Characteristic Lie algebra and Darboux integrable discrete chains

Differential-difference equation
$$ \frac d{dx}t(n+1,x)=f\left(x,t(n,x),t(n+1,x),\frac d{dx}t(n,x)\right) $$
with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$ is considered. An equation is said to be Darboux integrable, if there exist two functions (called integrals) $F$ and $I$ of a finite number of dynamical variables 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 $Dp(n)=p(n+1)$. It is proved that an equation is Darboux integrable if and only if its characteristic Lie algebras are finite-dimensional in both directions. The structure of integrals is described. Characteristic algebras for a certain class of integrable equations are described.