Rings
On the plane first (sphere should be similar)
Consider 3 concentric \(k\)-rings (rings with \(k\) identical vortices). The set of such configurations with centre 0 say is a sub-Hamiltonian system with 3 degrees of freedom (by symmetry arguments with cyclic group \(C_k\)). How similar is this to the 3-vortex system. For example, collinear configurations of 3 vortices should correspond to aligned or staggered rings. What about equilateral triangles?
Equilateral triangle
Geometry in curvature family
How does stability depend on \(\Gamma_j\) and curvature? How does \(\sigma_2\) enter? (vertical bifurcation in the plane - what about other geometries?)
Topology
In the plane with all \(\Gamma_j>0\):
Reduced space is \(CP^{n-2}\setminus\Delta\) (?). And \(H\to\infty\) at \(\Delta\), \(H\) is proper. What other critical points are there?
In the plane with \(\Gamma_j\) of mixed signs:
Hamiltonian is no longer proper - is it possible to get any critical points?
Representation Theory
On the sphere
Fix a (symmetric) configuration of vortices. The equation for an RE is a linear condition on the \(\Gamma_j\). The symmetry of the configuration acts on both the \(\Gamma_j\) and on \(\xi\in\mathfrak{g}\). That will suggest what the kernel must be.