Continuous nonpoint symmetries of ordinary difference equations

A method to derive the continuous nonpoint symmetries of ordinary difference equations (OΔE) of order two and higher is presented. A partial classification of second and fourth order difference equations that admit nonpoint symmetries both rational and polynomial forms which are quadratic in each variable is reported. Also, exploiting the obtained symmetries, it is shown how to construct integrals of motion or invariant for each of the considered equations. The question of integrability of the fourth order difference equations possessing the above type of nonpoint symmetries has also been briefly discussed.