On the first integrals of n-th order autonomous systems

The first integrals of n-th order autonomous systems are considered. By getting integrating factors using n−1 first integrals of the systems, a necessary condition on the existence of global first integrals of n-th order autonomous systems is presented. It is also proved that n−1 functionally independent first integrals of an n-th order autonomous system can be obtained if the system possesses n−1 functionally independent integrating factors. Based on one-parameter Lie groups admitted by n-th order autonomous systems, several methods to obtain first integrals of the systems are presented. Simultaneously, several related examples are given to illustrate the feasibility and the effectiveness of the proposed method.