Relative equilibria and Horn problem

The relative equilibria of $n$ bodies in $R^3$ submitted to the Newton (or similar) attraction are certainly the simplest possible solutions of the equations of motion. They exist only for very special configurations, the so-called central configurations whose determination is a very hard problem as soon as the number of bodies exceeds 3. The motions are periodic and necessarily take place in a fixed plane, a result which in the case of three bodies goes back to Lagrange.
Things become richer if one allows the dimension $d$ of the Euclidean ambient space to be greater than 3: then, a relative equilibrium is determined not only by its initial configuration but also by the choice of a hermitian structure on the space where the motion really takes place; moreover, for the so-called balanced configurations, more general than the central ones, the motion is in general quasi-periodic.
A way to distinguish up to isometry the relative equilibria of a given central (or balanced) configuration is to look for the frequency spectrum of their angular momentum bivector. Determining the set of these spectra, and in particular those for which a bifurcation from a periodic to a quasi-periodic relative equilibrium may occur by deformation of the configuration from central to balanced, is a purely algebraic problem whose solution is closely related to the classical problem of Horn.