  We study a many-body system of interacting spin-1 particles in the context of
homogeneous gases of ultracold atoms. In general, its description requires
eight parameters among which there are three components of magnetization and
five parameters associated with quadrupole degrees of freedom. Based on the
symmetry considerations, we construct a many-body interaction Hamiltonian that
includes eight generators of the SU(3) group related to the above description
parameters. The SU(3) symmetric Hamiltonian is applied to study the
ferromagnetic and quadrupolar phases of a homogeneous weakly interacting gas of
spin-1 atoms with Bose-Einstein condensate. It is shown how the quadrupole
degrees of freedom entering the Hamiltonian modify the ground state and
single-particle excitation spectra in comparison with those obtained from the
Hamiltonian bilinear in spin operators and not including quadrupole degrees of
freedom. We discuss the issue of taking into account the local character of
interaction to obtain the correct spectra of single-particle excitations.
