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

The work deals with the qualification of semidiscrete hyperbolic type equations. We study a class of equations of the form $$\frac{d{u}_{n+1}}{dx}=f(\frac{d{u}_n}{dx},u_{n+1},u_n),$$ here the unknown function $u_n(x)$ depends on one discrete $n$ and one continuous $x$ variables. Qualification is based on the requirement of the existence of higher symmetries. The case is considered when the symmetry is of order 5 in continuous directions. As a result, a list of four equations with the required conditions is obtained. For one of the found equations, a Lax representation is constructed.