  The purpose of this paper is twofold. First we study bifurcations of
connected sets of critical orbits of some invariant functional from a given
family of critical orbits. We use techniques of equivariant bifurcation theory
to obtain a Rabinowitz type alternative for symmetric gradient operators. The
second aim is to apply the abstract results to studying orbits of nonconstant
solutions of a nonlinear Neumann problem.
