On some exceptional cases in the integrability of the three-body problem

We consider the Newtonian planar three-body problem with positive masses m ₁, m ₂, m ₃. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian orbit besides three exceptional cases ∑m _i m _j /(∑m _k )² = 1/3, 2³/3³, 2/3² where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis of the three-body problem started in papers [Tsygvintsev, A.: Journal für die reine und angewandte Mathematik N 537, 127–149 (2001a); Celest. Mech. Dyn. Astron. 86(3), 237–247 (2003)] and based on the Morales–Ramis–Ziglin approach.