is $z$-component of the total spin operator, and $\gamma$ is the anisotropy parameter. The local spin operators $\sigma^\pm_n=(\sigma^x_n\pm i\sigma^y_n)/2$ and $\sigma^z_n$ are given by the Pauli matrices, which depend on the lattice argument $n\in\mathcal M\equiv\{1,2,\dots,M\}$, where $M=0\pmod{2}$. The corresponding commutation relations have the form: $$[\sigma^+_k,\sigma^-_l]\,=\,\delta_{k,l}\sigma^z_l\,,\quad
[\sigma^z_k,\sigma^\pm_l]\,=\,\pm2\delta_{k,l}\sigma^{\pm}_l\,.$$ The introduced *hopping matrix* $\Delta^{(s)}$ is defined by the following entries: $$\Delta^{(s)}_{nm}\equiv\frac12\,
(\delta_{|n-m|,1}+s\delta_{|n-m|,M-1}), \label{2}$$ where ${\delta}_{n,l}$ is the Kronecker symbol, and $s$ can take two values: $s=\pm$. It is assumed that the periodic boundary conditions $\sigma^{\alpha}_{n+M}=\sigma^{\alpha}_n$ are imposed for any $n\in\mathcal M$. The Hamiltonian $H$  is reduced to the Hamiltonian of the $XX$ magnet at zero value of the parameter $\gamma$.

The most general definition of the time $t$ and temperature $T\equiv
1/\beta$ dependent correlation functions of the model under consideration looks as follows: $$G_{j;l}^{ab}(t)\equiv\frac{1}Z
\Tr(\sigma^a_{j}(0)\sigma^b_l(t)e^{-\beta H}),\qquad
Z\equiv\Tr(e^{-\beta H}),
\label{3}$$ where $\sigma^b_l(t)\equiv e^{itH}\sigma^b_le^{-itH}$ and $\Tr$ means the averaging with respect to all eigen-states of the Hamiltonian $H$. In addition, the normalization involves the partition function $Z$. Calculation of the correlators  has been carried out in [@21] as averaging over all eigen-functions of the Hamiltonian of the $XX$ magnet. In [@21] the main attention has been paid to a relationship between the correlation functions and 