A note on the first integrals of the ABC system

作者：Llibre Jaume^*; Valls Claudia

来源：Journal of Mathematical Physics, 2012, 53(2): 023505.

DOI：10.1063/1.3682692

Without loss of generality the ABC systems reduce to two cases: either A = 0 and B, C %26gt;= 0, or A = 1 and 0 %26lt; B, C %26lt;= 1. In the first case it is known that the ABC system is completely integrable, here we provide its explicit first integrals. In the second case Ziglin [%26quot;Dichotomy of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom,%26quot; Izv. Akad. Nauk SSSR, Ser. Mat. 51, 1088 (1987)] proved that the ABC system with 0 %26lt; B %26lt; 1 and C %26gt; 0 sufficiently small has no real meromorphic first integrals. We improve Ziglin%26apos;s result showing that there are no C-1 first integrals under convenient assumptions.

• 出版日期2012-2