  We consider the system -\Delta u_j + a(x)u_j = \mu_j u_j^3 + \be\sum_{k\ne
j}u_k^2u_j, u_j>0, \qquad j=1,...,n, on a possibly unbounded domain
$\Om\subset\R^N$, $N\le3$, with Dirichlet boundary conditions. The system
appears in nonlinear optics and in the analysis of mixtures of Bose-Einstein
condensates. We consider the self-focussing (attractive self-interaction) case
$\mu_1,...,\mu_n > 0$ and take $\be\in\R$ as bifurcation parameter. There
exists a branch of positive solutions with $u_j/u_k$ being constant for all
$j,k\in{1,...,n}$. The main results are concerned with the bifurcation of
solutions from this branch. Using a hidden symmetry we are able to prove global
bifurcation even when the linearization has even-dimensional kernel (which is
always the case when $n>1$ is odd).
