A simplification and generalization of Elsayed's two-dimensional system of third order difference equations

A full Lie analysis of a system of third-order difference equations is performed. Explicit solutions, expressed in terms of the initial values, are derived. Furthermore, we give sufficient conditions for existence of 2-periodic and 4-periodic solutions in certain cases. Our results generalize and simplify the work by Elsayed and Ibrahim [E. M. Elsayed and T.F. Ibrahim, Periodicity and solutions for some systems of nonlinear rational difference equations, Hacettepe Journal of Mathematics and Statistics 44:6 (2015), 1361-1390].