A note on the first integrals of the ABC system

Abstract

Without loss of generality the ABC systems reduce to two cases: either A = 0 and B, C ⩾ 0, or A = 1 and 0 < B, C ⩽ 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 [“Dichotomy of the separatrices and the nonexistence of first integrals in systems of differential equations of Hamiltonian type with two degrees of freedom,” Izv. Akad. Nauk SSSR, Ser. Mat. 51, 1088 (1987)] proved that the ABC system with 0 < B < 1 and C > 0 sufficiently small has no real meromorphic first integrals. We improve Ziglin's result showing that there are no C1 first integrals under convenient assumptions.