  We consider a discrete equation, defined on the two-dimensional square
lattice, which is linearizable, namely, of the Burgers type and depends on a
parameter $\alpha$. For any natural number $N$ we choose $\alpha$ so that the
equation becomes Darboux integrable and the minimal orders of its first
integrals in both directions are greater or equal than $N$.

Exactly Solvable and Integrable Systems