Periodic families of solutions for difference equations with a first integral

In the following paper we establish that a one-parameter family of N- periodic solutions out of the origin is guaranteed to exist when the dimension of the N- periodic solution space of the corresponding linear problem is unity. When this dimension is greater than unity we establish that one parameter families generically exist. These results are obtained by adapting the method of Hale3 to a N-periodic difference equation with a N-periodic first integral