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

In this paper, a classification of semidiscrete equations of hyperbolic type is carried out. We study the class of equations of the form $$\frac{d{u}_{n+1}}{dx}=f(\frac{d{u}_n}{dx},u_{n+1},u_n),$$ here is the unknown function $u_n(x)$ depends on one discrete $n$ and one continuous $x$ variables. The classification is based on the requirement for the existence of higher symmetries in the discrete and continuous directions. The case is considered when the symmetries are of order 3 in both directions. As a result, a list of equations with the required conditions is obtained.