First integrals of discrete Birkhoffian system

For a discrete Birkhoffian system, symmetries and first integrals are presented. The operators of discrete transformation are introduced for the system. We extend the Noether-type identity in terms of the discrete Birkhoff's function. This identity makes it possible to establish the relationship between the symmetries and first integrals of discrete Birkhoffian system. Based on the discrete Pfaffian action, we give the discrete analog of symmetries of the system, and we provide a method to construct first integrals of discrete Birkhoffian equations. Finally, an example is given to illustrate the application of the method and the results show that the method is simple and effective.