A note on the first integrals of the ABC system

Abstract

Paper selected to be in <a href="http://jmp.aip.org/research_highlight_archive">Research Highlight Archive for 2012</A>

Agraïments: The second author has been partially supported by FCT through CAMGDS Lisbon.

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 in [14] 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.