Difference Krichever-Novikov operators

In this paper we study commuting difference operators of rank two. We introduce an equation on potentials $V(n),W(n)$ of the difference operator $L_4=(T+V(n)T^{-1})^2+W(n)$ and some additional data. With the help of this equation we find the first examples of commuting difference operators of rank two corresponding to spectral curves of higher genus.