notation we introduce $$\begin{aligned}
\label{bH}
\widehat{\bf H}_{\epsilon}(\eta,k)=\left[\begin{array}{cc}
-i\delta_{{\epsilon}}{\omega}(k;\eta)&-\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)\\
&\\
\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)&i\delta_{{\epsilon}}{\omega}(k;\eta)
\end{array}\right],\quad (\eta,k)\in{{\mathbb T}}_{{\epsilon}}\times {{\mathbb T}}\end{aligned}$$ and $$\begin{aligned}
\label{021605-19}
&&
{L}_\eta f(k):=2{\cal R}_{ \eta} f(k) - 2\bar R (k,\eta) f(k),\qquad
{L}_{\eta}^\pm f(k):=2{\cal R}_{\eta} f(k) -2R\left(k\pm\frac
   \eta2\right) f(k),\nonumber\\
&&
 {\cal R}_\eta f(k):=\int_{{{\mathbb T}}} R(k,k',\eta)f(k')dk',
\\
&&
R(k,k',\ell):=\frac12\sum_{\iota=\pm1}r\left(k-\frac{ \ell}{2},k-\iota
   k'\right)r\left(k+\frac{ \ell}{2},k-\iota k'\right),\quad
   k,k'\in{{\mathbb T}},\,\ell\in 2{{\mathbb T}}. \nonumber\end{aligned}$$ We denote by ${\frak
    L}_{{\epsilon}\eta}$, ${\frak H}_{\epsilon}$, ${\frak T}_{\epsilon}$ the operators, acting on ${\cal L}_{2,{\epsilon}}$, defined by $$\label{011906-19}
\widehat{{\frak
    L}_{{\epsilon}\eta}{\bf W}}=\widehat{\frak
    L}_{{\epsilon}\eta}\widehat{\bf W},\quad \widehat{{\frak
    H}_{{\epsilon}}{\bf W}}=\widehat{\frak
    H}_{{\epsilon}}\widehat{\bf W},\quad \widehat{{\frak
    T}_{{\epsilon}}{\bf W}}=\widehat{\frak
    T}_{{\epsilon}}\widehat{\bf W}.$$ Here $\widehat{\bf W}$ is the Fourier transform of ${\bf W}\in {\cal L}_{2,{\epsilon}}$, given by . Operator $\hat{\frak H}_{\epsilon}$, acting on $L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})$, is given by $$\begin{aligned}
\label{fH}
\widehat{\frak H}_{\epsilon}\widehat {{\bf W}}(\eta,k):=\widehat{\bf H}_{\epsilon}(\eta,k) \circ\widehat {{\bf W}}(\eta,k)=\left[\begin{array}{cc}
-i\delta_{{\epsilon}}{\omega}(k;\eta)\widehat W_{+}(\eta,k)&-\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)\widehat Y_{+}(\eta,k)\\
&\\
\dfrac{2i}{{\epsilon}} \bar{\omega}(k,{\epsilon}\eta)\widehat Y_{-}(\eta,k)&i\delta_{{\epsilon}}{\omega}(k;\eta)\widehat W_{-}(\eta,k)
\end{array}\right],\end{aligned}$$ with $\circ$ denoting the Hadamard’s product of $2\times 2$ matrices. Moreover, $\widehat{\frak
    L}_{{\epsilon}\eta}$ and $\widehat{\frak
    T}_{{\epsilon}}$ act on $L^2({{\mathbb T}}_{\epsilon}\times{{\mathbb T}})$ via the formulas $$\begin{aligned}
\label{LHT}
\widehat{\frak
    L}_{{\epsilon}\eta}\widehat {{\bf W}}(\eta,k):=\left[\begin{array}{cc}
\widehat W'_{+}(\eta,k)&\widehat Y'_{+}(\eta,k)\\
\widehat Y'_{-}(\eta,k)&\widehat W'_{-}(\eta,k)
\end{array}\right],\qquad \widehat{\frak
    T}_{{\epsilon}}\widehat {{\bf W}}(\eta,k):=\left[\begin{array}{cc}
\widehat W''_{+}(\eta,k)&\widehat Y''_{+}(\eta,k)\\
\widehat Y''_{-}(\eta,k)&\widehat W''_{-}(\eta,k)
\end{array}\right],\end{aligned}$$ with $$\begin{aligned}
\label{011605-19z}
&&\widehat W_{\pm}'(\eta,k)={L}_{{\epsilon}\eta}\widehat W_{\pm} (\eta,k)-\frac{1}{2}\sum_{\iota=\pm}{L}^{\pm}_{\iota{\epsilon}\eta}\widehat Y_{-\iota}(\eta,k),
\\
&&
\widehat Y_{\pm}'(\eta,k)={L}_{{\epsilon}\eta}\widehat Y_{\pm}(\eta,k)
+ {\cal R}_{{\epsilon}\eta}(\widehat
Y_{\mp}-\widehat 