A Note on the Relative Equilibria Bifurcations in the $$$$-Body Problem

Consider the planar Newtonian ((2N+1))-body problem, (Nge 1,) with (2N) bodies of unit mass and one body of mass (m). Using the discrete symmetry due to the equal masses and reducing by the rotational symmetry, we show that solutions with the (2N) unit mass points at the vertices of two concentric regular (N)-gons and (m) at the centre at all times form invariant manifold. We study the regular (2N)-gon with central mass (m) relative equilibria within the dynamics on the invariant manifold described above. As (m) varies, we identify the bifurcations, relate our results to previous work and provide the spectral picture of the linearization at the relative equilibria.