Hyperbolic equations with fifth-order symmetries

This paper examines the classification of hyperbolic equations. We study a class of equations of the form $$\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y}=F(\frac{\mathrm{\partial }u}{\mathrm{\partial }x},\frac{\mathrm{\partial }u}{\mathrm{\partial }y},u),$$ where $u(x,y)$ is the unknown function and $x,y$ are independent variables. The classification is based on the requirement for the existence of higher fifth-order symmetries. As a result, a list of four equations with the required conditions was obtained.