Classification of semidiscrete equations of hyperbolic type. The case of third-order symmetries

We classify semidiscrete equations of hyperbolic type. We study the class of equations of the form where the unknown function depends on one discrete ( ) and one continuous ( ) variables. The classification is based on the requirement that generalized symmetries exist in the discrete and continuous directions. We consider the case where the symmetries are of order in both directions. As a result, a list of equations with the required conditions is obtained.