  We obtain and justify a many-body Hamiltonian of pairwise interacting spin-1
atoms, which includes eight generators of the SU(3) group associated with spin
and quadrupole degrees of freedom. It is shown that this Hamiltonian is valid
for non-local interaction potential, whereas for local interaction specified by
$s$-wave scattering length, the Hamiltonian should be bilinear in spin
operators only (of the Heisenberg type). We apply the obtained Hamiltonian to
study the ground-state properties and single-particle excitations of a weakly
interacting gas of spin-1 atoms with Bose-Einstein condensate taking into
account the quadrupole degrees of freedom. It is shown that the system under
consideration can be in ferromagnetic, quadrupolar, and paramagnetic phases.
The corresponding phase diagram is constructed and discussed. The main
characteristics such as the density of the grand thermodynamic potential,
condensate density, and single-particle excitation spectra modified by
quadrupole degrees of freedom are determined in different phases.
