On the dihedral n-body problem

Consider n = 2l ≥ 4 point particles with equal masses in space, subject to the following symmetry constraint: at each instant they form an orbit of the dihedral group Dl, where Dl is the group of order 2l generated by two rotations of angle π around two secant lines in space meeting at an angle of π/l. By adding a homogeneous potential of degree −α for α in (0, 2) (which recovers the gravitational Newtonian potential), one finds a special n-body problem with three degrees of freedom, which is a kind of generalization of the Devaney isosceles problem, in which all orbits have zero angular momentum. In the paper we find all the central configurations and we compute the dimension of the stable/unstable manifolds