reduction to the system (\[N=3a\])-(\[N=3c\]), it consistently reduces when $$\mu=\bar{\mu},\quad  \rho=\bar{\rho}.$$ The relation $\rho=\bar{\rho}$ is satisfied if $$\begin{aligned}
&1)\, c_{23}a_1b_1=\bar{c}_{23}\bar{a}_1\bar{b}_1,\nonumber\\
&2)\, c_{23}a_1b_3+c_{23}a_3b_1=\bar{c}_{12}\bar{b}_1+\bar{c}_{13}\bar{a}_1+\bar{c}_{23}\bar{a}_1\bar{b}_2+\bar{c}_{23}\bar{a}_2\bar{b}_1,\nonumber\\
&3)\, c_{12}b_2+c_{13}a_2+c_{23}a_2b_2=\bar{c}_{23}\bar{a}_3\bar{b}_3,\nonumber\\
&4)\, c_{12}b_3+c_{13}a_3+c_{23}a_2b_3+c_{23}a_3b_2=\bar{c}_{12}\bar{b}_3+\bar{c}_{13}\bar{a}_3+\bar{c}_{23}\bar{a}_2\bar{b}_3+\bar{c}_{23}\bar{a}_3\bar{b}_2.\label{condlocalivN=3}\end{aligned}$$ The reduced equation is $$\label{localivN=3}
\mu v_{1,t}+6(\gamma_1v_1^2+\gamma_2v_1+\gamma_3\bar{v}_1^2+\gamma_4\bar{v}_1+\gamma_5v_1\bar{v}_1+\gamma_6)v_{1,x}+v_{1,xxx}=0,$$ where $$\begin{aligned}
&\gamma_1=c_{12}b_2+c_{13}a_2+c_{23}a_2b_2,\,  \gamma_2=c_{12}b_1+c_{13}a_1+c_{23}a_1b_2+c_{23}a_2b_1,\, \gamma_3=c_{23}a_3b_3 \\
&\gamma_4=c_{23}a_1b_3+c_{23}a_3b_1, \gamma_5=c_{12}b_3+c_{13}a_3+c_{23}a_3b_2+c_{23}a_2b_3,\, \gamma_6=c_{23}a_1b_1,\end{aligned}$$ with the conditions (\[condlocalivN=3\]) satisfied and $\mu \in \mathbb{R}$. When we use Type 1, we obtain the conditions on the parameters of one-soliton solution of the equation (\[localivN=3\]) as $$\begin{aligned}
&1)\, a_1=b_1=0,\,\,\, 2)\, k_1=k_2=k_3, k_i\in \mathbb{R}, i=1, 2, 3, \,\,\, 3)\, e^{\delta_2}=b_2e^{\delta_1}+b_3e^{\bar{\delta}_1},\nonumber \\
&4)\, e^{\delta_3}=a_2e^{\delta_1}+a_3e^{\bar{\delta}_1},\,\,\, 5)\, c_{ij}e^{\delta_i+\delta_j}=\bar{c}_{ij}e^{\bar{\delta}_i+\bar{\delta}_j}, i,j= 1, 2, 3.\end{aligned}$$ With the above conditions, the equation (\[localivN=3\]) becomes a cmKdV equation $$\label{reducedlocalivN=3}
\mu v_{1,t}+6(\gamma_1v_1^2+\gamma_3\bar{v}_1^2+\gamma_5|v_1|^2)v_{1,x}+v_{1,xxx}=0,$$ and one-soliton solution of (\[reducedlocalivN=3\]) is $$v_1=e^{k_1x-\frac{k_1^3}{\mu}t+\delta_1}.$$

### Nonlocal Reductions for N=3

**i.** $v_3(t,x)=a_3+a_2v_2(\varepsilon_1t,\varepsilon_2 x)+a_1v_1(\varepsilon_1t,\varepsilon_2 x)=a_3+a_2v_2^{\varepsilon}+a_1v_1^{\varepsilon}$, $\varepsilon_j^2=1$, $j=1, 2$, and $a_i$, $i=1, 2, 3$ are constants. When we use this reduction in (\[N=3b\]) for consistency of reduction we get the following conditions: $$1)\, \varepsilon_1\varepsilon_2=1,\quad 2)\, a_3=0,\quad 3)\, c_{12}=0,\quad 4)\, c_{13}a_2=c_{23}a_1.$$ Therefore to have a nonlocal equation, there is only one possibility; $(\varepsilon_1,\varepsilon_2)=(-1,-1)$. The reduced equation is ST-symmetric nonlocal mKdV system, $$\begin{aligned}
&\mu v_{1,t}(t,x)+6[c_{13}v_1(t,x)+c_{23}v_2(t,x)][a_1v_1(-t,-x)+a_2v_2(-t,-x)]v_{1,x}(t,x)+v_{1,xxx}(t,x)=0,\nonumber\\
&\mu v_{2,t}(t,x)+6[c_{13}v_1(t,x)+c_{23}v_2(t,x)][a_1v_1(-t,-x)+a_2v_2(-t,-x)]v_{2,x}(t,x)+v_{2,xxx}(t,x)=0.\label{nonlocaliN=3}\end{aligned}$$ If we use the Type 1 approach, the constraints that the parameters of the one-soliton solution of the 