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

Examples of Darboux Integrable Discrete Equations Possessing First Integrals of an Arbitrarily High Minimal Order