Symmetry, reductions and exact solutions of the difference equation $u_{n+2}=a{u}_n\mathrm{/}(1+b{u}_n{u}_{n+1})$

We investigate the solutions of the second-order difference equation $u_{n+2}=(a{u}_n)\mathrm{/}(1+b{u}_n{u}_{n+1})$ using a group of transformations (Lie symmetries) that leaves the solutions invariant.