---
abstract: |
    We study a many-body system of interacting spin-1 particles in the context of ultracold atomic gases. 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. This Hamiltonian can be generalized to spin-S systems by considering the generators of the SU(2S+1) group. We apply the Hamiltonian to study the ferromagnetic phase of a 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 containing the spin operators only. We discuss the issue of taking into account the local character of interaction to obtain the correct spectra of single-particle excitations.

    [**Keywords:**]{} high spin magnets; Ultracold gases, Bose-Einstein condensate; Ferromagnetic phase; Single-particle excitations
author:
- 'A.S. Peletminskii'
- 'S.V. Peletminskii'
- 'Yu.V. Slyusarenko'
title: 'SU(3) symmetry in theory of a weakly interacting gas of spin-1 atoms with Bose-Einstein condensate'
---

Introduction
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The magnetic properties 