is $\delta_1>0$ such that if $$U_0\in H^\infty\text{ and }\|U_0\|_{H^3}\leq \delta_1$$ then, for any $\varepsilon>0$, there is a solution $U=U^\varepsilon\in C([0,1]:H^{K+2})$ of the approximate equation $$\label{AppEq}
\left(\partial_t+i\Lambda-\varepsilon\Delta\right)U=\frac{i}{4}\sum_{j=1}^2\Lambda R_j\left[\vert\nabla\vert\Lambda^{-1}(U+\overline{U})\cdot R_j(U-\overline{U})\right]+\frac{i}{8}\sum_{j=1}^2\vert\nabla\vert\left[R_j(U-\overline{U})\cdot R_j(U-\overline{U})\right]$$ with initial data $U(0)=U_0$. Moreover, $$\label{ale1.1}
 \sup_{t\in[0,1]}\|U(t)\|_{H^3}\lesssim \delta_1$$ and, for any $\sigma\in[3,K]\cap\mathbb{Z}$ and $0\leq s\leq t\leq 1$, $$\label{ale1}
E_\sigma(t)\le E_\sigma(s)+C\int_s^t\|U(t')\|_{Z'}\cdot E_\sigma(t^\prime)dt^\prime,$$ where $$\label{DefE}
\begin{split}
&E_P:=\int_{\mathbb{R}^2}PU\cdot\overline{PU}dx-\frac{1}{8}\sum_{j=1}^2\int_{\mathbb{R}^2}\left[\vert\nabla\vert\Lambda^{-1}(U+\overline{U})\right]\cdot\left[R_jP(U-\overline{U})\cdot R_jP(U-\overline{U})\right]dx,\\
&E_\sigma:=\sum_{P=D^\alpha,\,\vert\alpha\vert\le \sigma}E_P.
\end{split}$$

By standard parabolic theory, for any fixed $\varepsilon>0$, the equation admits a local solution $U=U^\varepsilon\in C([0,T^\varepsilon],H^{K+2})$, for some $T^\varepsilon>0$, with $$\label{work}
\sup_{t\in[0,T^\varepsilon]}\|U(t)\|_{H^3}\lesssim \delta_1.$$ In view of the definition [^7], it follows that $$\label{work2}
\|U(t)\|_{H^\sigma}^2\lesssim E_{\sigma}(t)\lesssim \|U(t)\|_{H^\sigma}^2,\qquad t\in[0,T^\varepsilon],\,\sigma\in [0,K].$$

We rewrite $U=X+iY$, thus $$\label{realsystem}
\begin{split}
&\partial_tX=\Lambda Y+\varepsilon\Delta X-\sum_{j=1}^2\Lambda R_j(|\nabla|\Lambda^{-1}X\cdot R_jY),\\
&\partial_tY=-\Lambda X+\varepsilon\Delta Y-(1/2)\sum_{j=1}^2|\nabla|(R_jY\cdot R_jY),\\
&E_P=\int_{\mathbb{R}^2}(PX)^2+(PY)^2\,dx+\sum_{j=1}^2\int_{\mathbb{R}^2}|\nabla|\Lambda^{-1}X\cdot (R_jPY)^2\,dx.
\end{split}$$ Therefore, for $P=D^\alpha$, $|\alpha|\leq\sigma$, $$\frac{d}{dt}E_P=I_P+II_P+III_P+IV_P+V_P+VI_P-\varepsilon VII_P,$$ where $$I_P:=\int_{\mathbb{R}^2}-2PX\cdot \sum_{j=1}^2P\Lambda R_j(|\nabla|\Lambda^{-1}X\cdot R_jY)\,dx,$$ $$II_P:=\int_{\mathbb{R}^2}-PY\cdot \sum_{j=1}^2P|\nabla|(R_jY\cdot R_jY)\,dx,$$ $$III_P:=\sum_{j=1}^2\int_{\mathbb{R}^2}|\nabla|Y\cdot (R_jPY)^2\,dx,$$ $$IV_P:=\sum_{j=1}^2\int_{\mathbb{R}^2}-|\nabla|\Lambda^{-1}X\cdot 2R_jPY\cdot \Lambda R_jPX\,dx,$$ $$V_P:=\sum_{j=1}^2\int_{\mathbb{R}^2}-\sum_{m=1}^2|\nabla|R_m(|\nabla|\Lambda^{-1}X\cdot R_mY)\cdot (R_jPY)^2\,dx,$$ $$VI_P:=\sum_{j=1}^2\int_{\mathbb{R}^2}-|\nabla|\Lambda^{-1}X\cdot R_jPY\cdot R_jP|\nabla|\big(\sum_{m=1}^2R_mY\cdot R_mY\big)\,dx$$ $$\begin{split}
VII_P&:=2\int_{\mathbb{R}^2}\sum_{m=1}^2(\partial_mPX)^2+(\partial_mPY)^2\,dx\\
&+\sum_{j,m=1}^2\int_{\mathbb{R}^2}4\partial_m|\nabla|\Lambda^{-1}X\cdot R_jPY\cdot \partial_mR_jPY+2|\nabla|\Lambda^{-1}X\cdot (\partial_mR_jPY)^2\,dx.
\end{split}$$

We would like to bound the terms in the expression above in terms of $\|U\|_{H^\sigma}^2\|U\|_{Z'}$. For this we need the bounds $$\label{ka}
\begin{split}
&\sum_{m,j=1}^2\|\partial_jR_m U\|_{L^\infty}+\sum_{m,j=1}^2\|\partial_j(|\nabla|\Lambda^{-1}X\cdot R_mY)\|_{L^\infty}\lesssim \|U\|_{Z'},\\
&\|\Lambda(D^\alpha fD^\beta g)\|_{L^2}\lesssim\| \nabla f\|_{L^\infty}\Vert g\Vert_{H^{\sigma}}+\|\nabla g\|_{L^\infty}\| f\|_{H^{\sigma}},\qquad|\al|,|\be|\geq 1,\,|\alpha|+|\beta|\leq\sigma.
\end{split}$$ which are proved in Lemma \[tech1.5\] (recall that $\|U(t)\|_{H^3}\lesssim\delta_1$, $t\in[0,T^\varepsilon]$).

Using $$|III_P|+|V_P|\lesssim \|U\|_{H^\sigma}^2\|U\|_{Z'}.$$ Moreover, using 