be considered as the components of a single vector $q^{b}=(-\lambda^{1},-\lambda^{3},-\lambda^{4},-\lambda^{6},\lambda^{8})$. Therefore, if the microscopic constituents of a many-body system have a unit spin, then its macroscopic state is described by the eight parameters originating from the generators of the SU(3) group: $$\label{eq:2.9}
\Lambda^{a}=\sum_{\bf p}a^{\dagger}_{{\bf p}\alpha}\lambda^{a}_{\alpha\beta}a_{{\bf p}\beta}, \quad a=1,\ldots 8, \quad \alpha,\, \beta=x,\,y,\,z$$ that can be splitted into the spin and quadrupole operators, $$\label{eq:2.10}
S^{i}=\sum_{\bf p}a^{\dagger}_{{\bf p}\alpha}S^{i}_{\alpha\beta}a_{{\bf p}\beta} \quad (i=x,\,y,\,z), \quad Q^{b}=\sum_{\bf p}a^{\dagger}_{{\bf p}\alpha}q^{b}_{\alpha\beta}a_{{\bf p}\beta} \quad (b=1,\,3,\,4,\,6,\,8).$$

In the second quantization method the general two-body operator (or binary operator) $A^{(2)}$ can be expressed in the form (see, e.g., [@MethStatPhys]) $$A^{(2)}={1\over 4}\sum_{\alpha_{1},\ldots \alpha_{4}}a^{\dagger}_{\alpha_{1}}a^{\dagger}_{\alpha_{2}}A_{\alpha_{1}\alpha_{2};\alpha_{3}\alpha_{4}} a_{\alpha_{3}}a_{\alpha_{4}}, \quad A_{\alpha_{1}\alpha_{2};\alpha_{3}\alpha_{4}}=\langle \alpha_{1},\alpha_{2}|A^{(2)}|\alpha_{3},\alpha_{4}\rangle.$$ Therefore, one can write the two-body Hamiltonian describing the interaction of internal degrees of freedom in a many-body system of spin-1 particles, $$\label{eq:2.11}
V_{\lambda}={1\over 2\mathcal{V}}\sum_{{\bf p}_{1},\ldots{\bf p}_{4}}J({\bf p}_{1}-{\bf p}_{3})a^{\dagger}_{{\bf p}_{1}\alpha}a^{\dagger}_{{\bf p}_{2}\beta}\lambda^{a}_{\alpha \gamma}\lambda^{a}_{\beta\delta}a_{{\bf p}_{3}\gamma}a_{{\bf p}_{4}\delta}\,\delta_{{\bf p}_{1}+{\bf p}_{2},\,{\bf p}_{3}+{\bf p}_{4}},$$ where ${\cal V}$ is the volume of the system. The structure of the obtained Hamiltonian can also be justified within a phenomenological quasiparticle theory, where the energy of the system is considered to be a functional of the single-particle density matrix, like in the normal Fermi-liquid theory [@Landau; @Silin]. For a not dense system, one can restrict 