  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].
