Symmetry and bifurcations of planar configurations of the N-body and other problems

We describe a system of equations containing a real parameter β and an integer parameter N ≥2. Equilibria of these equations are in turn asymptotic shapes of systems of repelling particles for β= 0, central configurations with equal mass of the N-body problem for β= 1, and approximate solutions of a sphere-packing problem for β large. We introduce some asymmetric equilibria of these equations for N = 6 and 7, and identify and discuss the bifurcations that occur in this system for 5 ≤ N ≤ 8.