Abstract. We prove that the geometry of the 2-dimensional n-body problem for spaces of constant curvature κ = 0, n ≥ 3, does not allow for polygonal homographic solutions, provided that the corresponding orbits are irregular polygons of non-constant size.

It is well-known that the first and second Painlevé equations admit solutions characterised by divergent asymptotic expansions near infinity in specified sectors of the complex plane. Such solutions are pole-free in these sectors and called tronquée solutions by Boutroux. In this paper, we show that similar solutions exist for the third and fourth Painlevé equations as well.2010 Mathematics Subject Classification. Primary 33E17, 34M55.

We prove the existence of a class of rotopulsators for the n-body problem in spaces of constant curvature of dimension k ≥ 2.

We prove that if for the curved n-body problem the masses are given, the minimum distance between the point masses of a specific type of relative equilibrium solution to that problem has a universal lower bound that is not equal to zero. We furthermore prove that the set of all such relative equilibria is compact. This class of relative equilibria includes all relative equilibria of the curved n-body problem in S 2 , H 2 and a significant subset of the relative equilibria for