Phase portraits of the two-body problem with Manev potential

The Manev systems are two-body problems defined by a potential of the form a/r + b/r2, where r is the distance between the two particles, and a and b are arbitrary constants. The Hamiltonian H = (p2r + p2θ/r2)/2 + a/r + b/r2 and the angular momentum pθ = r2θ̇ associated with Manev systems are two first integrals, which are independent and in involution. Let Ih (respectively Ic) be the set of points of the phase space on which H (respectively pθ) takes the value h (respectively c). Since H and pθ are first integrals, the sets Ih, Ic and Ihc = Ih ∩ Ic are invariant under the flow of the Manev systems. We characterize the global flow of these systems when a and b vary. Thus we describe the foliation of the phase space by the invariant sets Ih and the foliation of Ih by the invariant sets Ihc.