eight consistent reductions; four of them are local and the others are nonlocal. To obtain one-soliton solution of the reduced equations one can use Type 1 and Type 2 approaches given in [@GurPek2]. Here we will only use Type 1.

### Local Reductions for N=3

We have four different local reductions for (\[N=3a\])-(\[N=3c\]).

**i.** $v_3=a_3+a_2v_2+a_1v_1$, $a_i$, $i=1, 2, 3$ are constants. When we use this reduction, the system (\[N=3a\])-(\[N=3c\]) consistently reduces to $$\begin{aligned}
&\mu v_{1,t}+6((c_{12}+c_{13}a_2+c_{23}a_1)v_1v_2+c_{13}a_1v_1^2+c_{23}a_2v_2^2+c_{13}a_3v_1+c_{23}a_3v_2)v_{1,x}+v_{1,xxx}=0,\nonumber\\
&\mu v_{2,t}+6((c_{12}+c_{13}a_2+c_{23}a_1) v_1v_2+c_{13}a_1v_1^2+c_{23}a_2v_2^2+c_{13}a_3v_1+c_{23}a_3v_2)v_{2,x}+v_{2,xxx}=0, \label{localiN=3}\end{aligned}$$ without any additional condition. If we follow the Type 1 approach, the constraints that the parameters of the one-soliton solution of the system (\[localiN=3\]) are obtained from $$\frac{g_3}{f}=a_3+a_2\frac{g_2}{f}+a_1\frac{g_1}{f} \Rightarrow g_3=a_3f+a_2g_2+a_1g_1$$ as $$1)\, a_3=0,\quad 2)\, k_1=k_2=k_3,\quad 3)\, e^{\delta_3}=a_2e^{\delta_2}+a_1e^{\delta_1}.$$ In this case we have $v_2=\alpha v_1$, where $\alpha=e^{\delta_2-\delta_1}$, and the system (\[localiN=3\]) becomes a single equation, usual mKdV equation, $$\label{reducedeqlocaliN=3}
\mu v_{1,t}+6\zeta v_1^2 v_{1,x}+v_{1,xxx}=0,$$ where $\zeta=(c_{12}+c_{13}a_2+c_{23}a_1)\alpha+c_{13}a_1+c_{23}a_2\alpha^2$. We obtain the one-soliton solution of (\[reducedeqlocaliN=3\]) as $$\displaystyle
v_1=\frac{e^{k_1x-\frac{k_1^3}{\mu}t+\delta_1}}{1+\frac{1}{4k_1^2}[c_{13}a_1e^{2\delta_1}+c_{23}a_2e^{2\delta_2}
+(c_{12}+c_{13}a_2+c_{23}a_1)e^{\delta_1+\delta_2}]e^{2k_1x-\frac{2k_1^3}{\mu}t}}.$$

**ii.** $v_3=a_3+a_2\bar{v}_2+a_1\bar{v}_1$, $a_i$, $i=1, 2, 3$ are constants. If we apply this reduction to the system (\[N=3a\])-(\[N=3c\]), it consistently reduces if $$\mu=\bar{\mu},\quad  \rho=\bar{\rho}.$$ The relation $\rho=\bar{\rho}$ is satisfied if $$\label{condlocaliN=3}
1)\, c_{12}=0,\, 2)\, a_3=0,\, 3)\, c_{13}a_2=\bar{c}_{23}\bar{a}_1,\, 4)\, c_{13}a_1=\bar{c}_{13}\bar{a}_1,\,
5)\,c_{23}a_2=\bar{c}_{23}\bar{a}_2.$$ The reduced system is $$\begin{aligned}
&\mu v_{1,t}+6(c_{13}a_2 