  We study existence and non-existence of constant scalar curvature metrics
conformal and arbitrarily close to homogeneous metrics on spheres, using
variational techniques. This describes all critical points of the
Hilbert-Einstein functional on such conformal classes, near homogeneous
metrics. Both bifurcation and local rigidity type phenomena are obtained for
1-parameter families of U(n+1), Sp(n+1) and Spin(9)-homogeneous metrics.
