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

We describe a system of equations containing a real parameter (beta) and an integer parameter N > 2. Equilibria of these equations are in turn asymptotic shapes of systems of repelling particles for(beta) = 0, central configurations with equal mass of the N-body problem for(beta) = I, and approximate solutions of a sphere-packing problem for(beta) 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