  We study the stability and bifurcation of relative equilibria of a particle
on the Lie group $SO(3)$ whose motion is governed by an $SO(3)\times SO(2)$
invariant metric and an $SO(2)\times SO(2)$ invariant potential. Our method is
to reduce the number of degrees of freedom at singular values of the
$SO(2)\times SO(2)$ momentum map and study the stability of the equilibria of
the reduced systems as a function of spin. The result is an elementary analysis
of the fast/slow transition in the Lagrange and Kirchhoff tops.
  More generally, since an $SO(2)\times SO(2)$ invariant potential on $SO(3)$
can be thought of as ${\mathbb Z}_2$ invariant function on a circle, we analyze
the stability and bifurcation of relative equilibria of the system in terms of
the second and fourth derivative of the function.
