On Miura transformations and Volterra-type equations

We construct Miura transformations mapping the scalarspectral problems of the integrable lattice equations belonging tothe Adler-Bobenko-Suris (ABS) list into the discrete Schrodingerspectral problem associated with Volterra-type equations. We showthat the ABS equations correspond to Backlund transformations forsome particular cases of the discrete Krichever-Novikov equationfound by Yamilov (YdKN equation). This enables us to construct newgeneralized symmetries for the ABS equations. The same can be saidabout the generalizations of the ABS equations introduced by Tongas,Tsoubelis and Xenitidis. All of them generate Backlundtransformations for the YdKN equation. The higher order generalizedsymmetries we construct in the present paper confirm theirintegrability.