  We consider the task of classifying relative equilibria for mechanical
systems with rotational symmetry. We divide relative equilibria into two
natural groups: a generic class which we call normal, and a non-generic
abnormal class. The eigenvalues of the locked inertia tensor descend to
shape-space and endow it with the geometric structure of a 3-web with the
property that any normal relative equilibrium occurs as a critical point of the
potential restricted to a leaf from the web. To demonstrate the utility of this
web structure we show how the spherical 3-body problem gives rise to a web of
Cayley cubics on the 3-sphere, and use this to fully classify the relative
equilibria for the case of equal masses.
