On the meromorphic non-integrability of some $N$-body problems

We present a proof of the meromorphic non--integrability of the planar $N$-Body Problem for some special cases. A simpler proof is added to those already existing for the Three-Body Problem with arbitrary masses. The $N$-Body Problem with equal masses is also proven non-integrable. Furthermore, a new general result on additional integrals is obtained which, applied to these specific cases, proves the non-existence of an additional integral for the general Three-Body Problem, and provides for an upper bound on the amount of additional integrals for the equal-mass Problem for $N=4,5,6$. These results appear to qualify differential Galois theory, and especially a new incipient theory stemming from it, as an amenable setting for the detection of obstructions to Hamiltonian integrability.