  We consider a small SO(2)-equivariant perturbation of a reaction-diffusion
system on the sphere, which is equivariant with respect to the group SO(3) of
all rigid rotations. We consider a normally hyperbolic SO(3)-group orbit of a
rotating wave on the sphere that persists to a normally hyperbolic
SO(2)-invariant manifold $M(\epsilon)$. We investigate the effects of this
forced symmetry breaking by studying the perturbed dynamics induced on
$M(\epsilon)$ by the above reaction-diffusion system. We prove that depending
on the frequency vectors of the rotating waves that form the relative
equilibrium SO(3)u_{0}, these rotating waves will give SO(2)-orbits of rotating
waves or SO(2)-orbits of modulated rotating waves (if some transversality
conditions hold). The orbital stability of these solutions is established as
well. Our main tools are the orbit space reduction, Poincare map and implicit
function theorem.
