Topology of the Generalized Constrained Three-Vortex Problem at Zero Total Vortical Moment

We consider a completely Liouville integrable Hamiltonian system with two degrees of freedom. The system describes the dynamics of three magnetic vortices with a constraint imposed on one of the vortices. For positive parameters of relative polarities, the system describes the motion of two unconstrained vortex filaments in an unbounded perfect fluid in the field of a fixed vortex. This paper studies the topology of the system’s Liouville foliation for zero total vortical moment. Having changed the parameters, we consider two one-parameter families of Liouville integrable Hamiltonian systems: one corresponds to equal relative vortex polarities, the other to opposite ones. We study the topology of the noncompact isoenergy 3-manifolds and their bifurcations through singular isoenergy surfaces. Then, we construct all the augmented bifurcation diagrams of the momentum map that are possible in the case under consideration and provide a fiberwise description for the bifurcations of the integral 2-manifolds. Among the bifurcations there are noncompact and noncritical ones. In particular, we found a new unstable noncompact bifurcation formed by one critical and one noncritical bifurcation placed on the same bifurcation fiber. This bifurcation is of complexity \(3/2\) and has two different splittings under small integrable parameter perturbations.