the ordinary magnetization vector. These parameters are related to the quadrupole degrees of freedom. From the symmetry considerations, we have found the many-body SU(3) symmetric Hamiltonian describing the interaction of internal degrees of freedom. It includes eight generators of the SU(3) group, among which there are three spin and five quadrupole operators. The Hamiltonian has been applied to study the ferromagnetic state of a weakly interacting Bose gas of spin-1 atoms with Bose-Einstein condensate. It has been shown that the ground state is specified by non-zero magnetization vector and additional quadrupole parameter. In addition, due to the quadrupole degrees of freedom entering the Hamiltonian, the single-particle excitation spectra are modified in comparison with those obtained from the usually employed Hamiltonian containing only the spin operators. It is also worth noting that the nonlocal character of interaction essentially affects the structure of the spectra so that the parametrization of interaction by the scattering length is a sufficiently rough approximation. Finally, the studied Hamiltonian can be easily generalized to spin-S systems by considering the generators $T^{a}$ of the SU(N) group with $N=2S+1$. The resulting Hamiltonian can be applied to study the magnetic properties of high-spin dilute Fermi and Bose gases.

The properties 