  We study bifurcation for the constant scalar curvature equation along a
one-parameter family of Riemannian metrics on the total space of a harmonic
Riemannian submersion. We provide an existence theorem for bifurcation points
and a criterion to see that the conformal factors corresponding to the
bifurcated metrics must be indeed constant along the fibers. In the case of the
canonical variation of a Riemannian submersion with totally geodesic fibers, we
characterize discreteness of the set of all degeneracy points along the family
and give a sufficient condition to guarantee that bifurcation necessarily
occurs at every point where the linearized equation has a nontrivial solution.
In the model case of quaternionic Hopf fibrations, we show that
symmetry-breaking bifurcation does not occur except at the round metric.
