Symmetries and First Integrals of Differential Equations

It is known that an n-dimensional system of ordinary differential equations with Lie symmetry which involves a divergence-free Liouville vector field possesses n−1 independent first integrals (i.e., it is algebraically integrable) (ünal in Phys. Lett. A 260:352–359, 1999]). In the present paper, we show that if an n-dimensional system of ordinary differential equations admits a C ^∞-symmetry vector field which satisfies some special conditions, then it also possesses n−1 independent first integrals. Several examples are given to illustrate our result. Y. Li’s research was partially supported by NSFC Grants 10531050, 10225107, SRFDP Grant 20040183030, and the 985 program of Jilin University.