Generic symmetry-reduced integrability conjecture for point-vortex Hamiltonian systems

Let $M$ be a 2-dimensional Riemannian manifold with symmetry group $G$. Let $P^N_M$ denote the phase space of $N$ point vortices on $M$, and consider a generic $G$-invariant Hamiltonian system on this space. The relevant geometric conditions are that the action of $G$ is free and proper and that the momentum map is equivariant.

Generic symmetry-reduced integrability conjecture. A generic $G$-invariant Hamiltonian system on $P^N_M$ is integrable only when these geometric conditions are such that symplectic reduction directly implies integrability.

This conjecture summarizes the survey's evidence from point-vortex dynamics: quasi-periodic or relative-equilibrium trajectories occur when the hypotheses of the symplectic-reduction integrability theorem hold, whereas violations such as non-vanishing momentum or circulation are associated with chaotic trajectories. The parser supplies no evidence that the conjecture has been resolved.