allows to reduce the general Hermitian quadratic form in bosonic operators to a diagonal structure [@BogBog]. Therefore, let us introduce the unitary operator $U$ mixing up $a_{{\bf p}\lambda}$ and $a^{\dagger}_{{\bf p}-\lambda}$: $$\begin{gathered}
Ua_{{\bf p}\alpha}U^{\dagger}=\sum_{\lambda=x,y}\left[u_{\alpha\lambda}({\bf p})a_{{\bf p}\lambda}+v^{*}_{\alpha\lambda}({\bf p})a^{\dagger}_{-{\bf p}\lambda}\right], \nonumber \\
Ua^{\dagger}_{{\bf p}\alpha}U^{\dagger}=\sum_{\lambda=x,y}\left[u^{*}_{\alpha\lambda}({\bf p })a^{\dagger}_{{\bf p}\lambda}+v_{\alpha\lambda}({\bf p})a_{-{\bf p}\lambda}\right] \label{eq:4.14}\end{gathered}$$ and transforming ${\cal H}_{2}^{(2)}(n_{0})$ to the diagonal form $$\label{eq:4.15}
U{\cal H}_{2}^{(2)}(n_{0})U^{\dagger}=\sum_{{\bf p}\neq 0}\sum_{\lambda=x,y}\omega_{{\bf p}\lambda}a^{\dagger}_{{\bf p }\lambda}a_{{\bf p}\lambda}+{\cal E}_{0},$$ where $\omega_{{\bf p}\lambda}$ are the spectra of single-particle excitations and ${\cal E}_{0}$ redefines the vacuum energy or the ground state thermodynamic potential. The creation and annihilation operators $Ua^{\dagger}_{{\bf p}\alpha}U^{\dagger}$ and $Ua_{{\bf p}\alpha}U^{\dagger}$ given by Eqs. (\[eq:4.14\]) must satisfy the same bosonic commutation relations as the operators $a_{{\bf p}\alpha}$ and $a^{\dagger}_{{\bf p}\alpha}$. This requirement results in the following normalization and orthogonality conditions for functions $u_{\alpha\lambda}({\bf p})$ and $v_{\alpha\lambda}(\bf p)$: $$\begin{gathered}
\sum_{\lambda=x,y}\left[u_{\alpha\lambda}({\bf p})u^{*}_{\beta\lambda}({\bf p})-v^{*}_{\alpha\lambda}({\bf p})v_{\beta\lambda}({\bf p})\right]=\delta_{\alpha\beta}, \nonumber \\
\sum_{\lambda=x,y}\left[u_{\alpha\lambda}({\bf p})v^{*}_{\beta\lambda}({\bf p})-v^{*}_{\alpha\lambda}({\bf p})u_{\beta\lambda}({\bf p})\right]=0. \label{eq:4.16}\end{gathered}$$ Note that $u({\bf p})$ and $v({\bf p})$ are constructed from the same quantities as the matrices $A_{\alpha\beta}({\bf p})$ and $B_{\alpha\beta}({\bf p})$ and, therefore, they can be considered as even functions of momentum. The energies of single-particle excitations (or quasiparticles) $\omega_{{\bf p}\lambda}$ satisfy the following eigenvalue equations [@BogBog]: $$\begin{gathered}
\sum_{\lambda=x,y}\left[A_{\alpha\lambda}({\bf p })u_{\lambda\gamma}({\bf p 