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
$$\frac{du_{n+1}}{dx}=f\biggl(\frac{du_{n}}{dx},u_{n+1},u_{n}\biggr),$$
where the unknown function $u_n(x)$ depends on one discrete ($n$) and one continuous ($x$) 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 $3$ in both directions. As a result, a list of equations with the required conditions is obtained.