On solvability of a two-dimensional symmetric nonlinear system of difference equations

We show that the system of difference equations xₙ₊ₖ=xₙ₊ₗyₙ-efxₙ₊ₗ+yₙ-e-f, yₙ₊ₖ= yₙ₊ₗxₙ-efyₙ₊ₗ+xₙ-e-f, n∈ N₀, where k∈ N , l∈ N₀ , $l< k$ , e, f∈ C , and xⱼ, yⱼ∈ C , j=0,k-1̄ , is theoretically solvable and present some cases of the system when the general solutions can be found in a closed form.