Non integrability of the n body problem with non zero angular momentum

We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree -1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n body problem with Newtonian interaction in the plane on a surface of equation $(H,C)=(H_0,C_0)$ with $(H_0,C_0) \neq (0,0)$ where C is the angular momentum and H the energy, in the case where the n masses are equal.

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