ourselves by the energy functional quadratic in the single-particle density matrix [@FNT]. Since each density matrix represents the eight-component vector, they should enter the energy functional as a scalar product. On the other hand, it is clear, that such a functional should be obtained from the microscopic Hamiltonian given by Eq. (\[eq:2.11\]) by statistical averaging and using the Bloch–De Dominicis (or Wick’s) theorem [@BogBog].

Taking into account the commutation relations for the creation and annihilation operators as well as the properties of the structure constants $f^{abc}$ (see Eqs. (\[eq:2.1\]), (\[eq:A4\])), one can show that $[V_{\lambda},\Lambda^{a}]=0$ and, consequently, $\Lambda_{a}$ is the integral of motion. Note that the above Hamiltonian and the corresponding macroscopic state are SU(3) symmetric. This symmetry, however, is broken if the coupling between the spin and magnetic field (the Zeeman term) is taken into account. The role of SU(3) symmetry has been discussed when studying the dynamics and relaxation in high-spin magnets [@Kov2; @Kov3; @Ivanov].

Truncated Hamiltonian for a weakly interacting Bose gas with internal degrees of freedom
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As we noted, ultracold quantum gases provide a powerful tool to study various effects and phenomena in quantum many-body systems. Therefore, we apply the obtained Hamiltonian to study the 