  We propose a method for systematically finding ground states of spinor
Bose-Einstein condensates by utilizing symmetry properties of the system. By
this method, we can find not only an inert state, whose symmetry is maximal in
the manifold under consideration, but also a non-inert state, which has lower
symmetry and depends on the parameters in the Hamiltonian. We establish the
symmetry-classification method for the spin-1, 2 and 3 cases at zero magnetic
field, and find a new phase in the last case. Properties of vortices in the
spin-3 system are also discussed.
