Reductions, relative equilibria, and bifurcations in the generalized van der Waals potential: Relation to the integrable cases

Complementing the work of Alhassid et al. we study the global dynamics of the averaged system of the generalized van der Waals interaction in the reduced space which is a two-dimensional sphere. We find lines of local ``pitchfork'' and global ``oyster'' bifurcations emerging from the known integrable cases {β}=1/2, 1, 2; this explains the chaos-order-chaos transition. We present the libration and circulation modes of the Runge-Lenz vector and its stability domains. The appearance-disappearance pattern of separatrices for the known integrable cases leads us to conjecture that those are the only ones.