  The aim of this paper is to study global bifurcations of non-constant
solutions of some nonlinear elliptic systems, namely the system on a sphere and
the Neumann problem on a ball. We study the bifurcation phenomenon from
families of constant solutions given by critical points of the potentials.
Considering this problem in the presence of additional symmetries of a compact
Lie group, we study orbits of solutions and, in particular, we do not require
the critical points to be isolated. Moreover, we allow the considered orbits of
critical points to be degenerate. To prove the bifurcation we compute the index
of an isolated degenerate critical orbit in an abstract situation. This index
is given in terms of the degree for equivariant gradient maps.
