  The purpose of this paper is to study properties of continua (closed
connected sets) of nontrivial solutions of non-cooperative elliptic systems
considered on geodesic balls in $S^n$. In particular, we have shown that if the
geodesic ball is a hemisphere, then these continua are unbounded. It is also
shown that the phenomenon of global symmetry-breaking bifurcation of such
solutions occurs. Since the problem is variational and SO(n)-symmetric, we
apply the techniques of equivariant bifurcation theory to prove the main
results of this article. As the topological tool we use the degree theory for
SO(n)-invariant strongly indefinite functionals defined in A.
Go{\l}\c{e}biewska, S. Rybicki, \emph{Global bifurcations of critical orbits of
$G$-invariant strongly indefinite functionals}, Nonl. Anal. {74} (2011),
1823-1834..
