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.
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