Necessary integrability conditions for evolutionary lattice equations

Abstract: The structure of solutions is studied for the Lax equation $D_t(G)=[F,G]$ for formal power series with respect to the shift operator. It is proved that if the equation with a given series $F$ of degree $m$ admits a solution $G$ of degree $k$ then it admits, as well, a solution $H$ of degree $m$ such that $H^k=G^m$. This property is used for derivation of necessary integrability conditions for scalar evolutionary lattices.