symmetric approach adopted here, all these averages can be expressed in terms of the two-site spin correlation functions $C_{{\bf r}}=\langle {\bf 
S}_0\cdot {\bf S}_{{\bf r} }\rangle $. Let us recall that due to the spherical symmetry $\left\langle S_i^\alpha S_j^\beta 
\right\rangle =\delta _{\alpha \beta }\left\langle S_i^\alpha 
S_j^\alpha \right\rangle =\frac 13\left\langle {\bf S}_i\cdot 
{\bf S}_j\right\rangle $, $\left\langle S_i^\alpha \right\rangle =0$. Simultaneously, at $T\rightarrow 0$ the spin subsystem is described by a long range order state with finite effective magnetization $m$, $C_{{\bf r}
}(|{\bf r|}\rightarrow \infty )=m^2(-1)^{r_x+r_y}$, the value of $m$ is dictated by the Bose condensation of spin excitations at the antiferromagnetic vector ${\bf q}_0=(\pi ,\pi )$.

Note, that ${\cal B}_{{\bf k},\sigma }^{\dagger }|L\rangle $ corresponds to the CuO$_2$ plane state with the total spin equal to $\frac 12$ if $
|L\rangle $ is the singlet state. We treat ${\cal B}_{k,\sigma }^{\dagger }$ as a candidate for the elementary excitations operator and calculate the corresponding two-time retarded Green’s function $G({\bf k},\omega )$ and spectral density $$A_p({\bf k},\omega )=-\frac 1\pi {\rm Im}G({\bf k},\omega +\imath 0^{+}),$$ $$\label{pGf}G({\bf k},\omega )=\langle {\cal B}_{{\bf k},\sigma }|{\cal B}_{
{\bf k},\sigma }^{\dagger }\rangle _\omega \equiv -\imath \int_{t^{\prime
}}^\infty \!\!dte^{\imath \omega (t-t^{\prime })}\langle \{{\cal B}_{{\bf k}
,\sigma }(t),{\cal B}_{{\bf k},\sigma 