v_1\bar{v}_2+c_{23}a_1\bar{v}_1v_2+c_{13}a_1|v_1|^2+c_{23}a_2|v_2|^2)v_{1,x}+v_{1,xxx}=0,\nonumber\\
&\mu v_{2,t}+6(c_{13}a_2 v_1\bar{v}_2+c_{23}a_1\bar{v}_1v_2+c_{13}a_1|v_1|^2+c_{23}a_2|v_2|^2)v_{2,x}+v_{2,xxx}=0,\label{localiiN=3}\end{aligned}$$ with the conditions (\[condlocaliN=3\]) satisfied and $\mu \in \mathbb{R}$. If we use the Type 1 approach to find one-soliton solution of the above system, we obtain the constraints that the parameters of the one-soliton solution of the system (\[localiiN=3\]) from $$\frac{g_3}{f}=a_2\frac{\bar{g}_2}{\bar{f}}+a_1\frac{\bar{g}_1}{\bar{f}} \Rightarrow f=\bar{f}, \, g_3=a_2\bar{g}_2+a_1\bar{g}_1,$$ as $$1)\, k_3=\bar{k}_2=\bar{k}_1,\quad 2)\, e^{\delta_3}=a_2e^{\bar{\delta}_2}+a_1e^{\bar{\delta}_1}.$$ In this case we have $v_2=e^{\delta_2-\delta_1} v_1$ and the system (\[localiiN=3\]) becomes a single cmKdV equation, $$\label{reducedeqlocaliiN=3}
\mu v_{1,t}+6[c_{13}(\bar{\alpha}a_2+a_1)+c_{23}(\alpha a_1+\alpha^2 a_2)]|v_1|^2 v_{1,x}+v_{1,xxx}=0.$$ One-soliton solution of the equation (\[reducedeqlocaliiN=3\]) is $$\displaystyle
v_1=\frac{e^{k_1x-\frac{k_1^3}{\mu}t+\delta_1}}{1+\frac{1}{(k_1+\bar{k}_1)^2}(c_{13}e^{\delta_1}+c_{23}e^{\delta_2})(a_2e^{\bar{\delta}_2}
+a_1e^{\bar{\delta}_1})e^{(k_1+\bar{k}_1)x-\frac{(k_1^3+\bar{k}_1^3)}{\mu}t}}.$$

**iii.** $v_3=a_1+a_2v_1$, $v_2=b_1+b_2v_1$ $a_i, b_i$, $i=1, 2, 3$ are constants. When we apply this reduction to the system (\[N=3a\])-(\[N=3c\]) it consistently reduces to the equation $$\label{localiiiN=3}
\mu v_{1,t}+6([c_{12}b_2+c_{13}a_2+c_{23}a_2b_2]v_1^2+[c_{12}b_1+c_{13}a_1+c_{23}a_1b_2+c_{23}a_2b_1]v_1+c_{23}a_1b_1)v_{1,x}+v_{1,xxx}=0,$$ without any additional condition. When we use the Type 1 approach, we obtain the constraints that the parameters of the one-soliton solution of the equation (\[localiiiN=3\]) are obtained from $$\frac{g_3}{f}=a_1+a_2\frac{g_1}{f},\, \frac{g_2}{f}=b_1+b_2\frac{g_1}{f}  \Rightarrow g_3=a_1f+a_2g_1,\, g_2=b_1f+b_2g_1$$ as $$1)\, a_1=b_1=0,\, 2)\, k_1=k_2=k_3,\, 3)\, e^{\delta_3}=a_2e^{\delta_1},\, 4)\, e^{\delta_2}=b_2e^{\delta_1},
\, 5)\, c_{12}=-a_2c_{23}, \, 6)\, c_{13}=-b_2c_{23}.$$ In this case the equation (\[localiiiN=3\]) becomes the usual mKdV equation, $$\label{reducedeqlocaliiiN=3}
\mu v_{1,t}+6c_{12}b_2 v_1^2 v_{1,x}+v_{1,xxx}=0,$$ and its one-soliton solution is $$\displaystyle
v_1=\frac{e^{k_1x-\frac{k_1^3}{\mu}t+\delta_1}}{1+\frac{e^{2\delta_1}}{4k_1^2}c_{12}b_2e^{2k_1x-\frac{2k_1^3}{\mu}t}}.$$

**iv.** $v_3=a_1+a_2v_1+a_3\bar{v}_1$, $v_2=b_1+b_2v_1+b_3\bar{v}_1$ $a_i, b_i$, $i=1, 2, 3$ are constants. If we apply this 