Examples of Darboux Integrable Discrete Equations Possessing First   Integrals of an Arbitrarily High Minimal Order
  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 $N$. 
Rustem N. Garifullin and Ravil I. Yamilov
nlin.SI