We use a recently found method to characterise all the invertible fourth-order difference equations linear in the extremal values based on the existence of a discrete Lagrangian. We also give some result on the integrability properties of the obtained family and we put it in relation with known classifications. Finally, we discuss the continuum limits of the integrable cases.
Gubbiotti, G. (2020). Lagrangians and integrability for additive fourth-order difference equations. THE EUROPEAN PHYSICAL JOURNAL PLUS, 135(10) [10.1140/epjp/s13360-020-00858-y].