solution obtained by Type 1 approach [@GurPek2], they reduce to a single known equation.

Local Reductions
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We define new sets of indices. Let the Greek indices $\alpha, \beta, \gamma, \ldots$ run from 1 to $M$ and let the Latin alphabet $a,b,c,\ldots $ runs from $M+1$ to $N$. Then $v_{i}=(v_{\alpha}, v_{a})$ and we can write $ \rho$ defined in (\[rho\]) as

$$\rho=c_{\alpha \beta}\, v_{\alpha}\, v_{\beta}+ 2c_{\alpha a}\, v_{\alpha} v_{a}+c_{ab}\, v_{a}\,v_{b},$$

where $c_{jk}=c_{kj}$, $c_{jj}=0$, and repeated indices are summed up over their range. There are two types of local reductions.

[**(a)**]{} Let $v_{a}=k_{a}+A_{a \alpha}\,v_{\alpha}$ then the system (\[coupledmKdVN\]) consistently reduces to

$$\label{eqn1}
\mu v_{\alpha,t}+3 \rho \,v_{\alpha,x}+v_{\alpha,xxx}=0,~~\alpha=1,2,\ldots, M,$$

where $k_{a}$ and $A_{a \alpha}$ are arbitrary constants and $$\rho=q_{\alpha \beta}\, v_{\alpha} \, v_{\beta}+q_{\alpha} \, v_{\alpha}+q,$$ where $$\begin{aligned}
q&=&c_{ab}\, k_{a} k_{b}, \\
q_{\alpha}&=& 2k_{a}\, c_{a \alpha}+2k_{a}\, c_{ab} A_{b \alpha},~  \alpha=1,2,\ldots, M,\\
q_{\alpha \beta}&=&c_{\alpha \beta}+c_{a \alpha}\, A_{a \beta}+c_{a \beta}\, A_{a \alpha}+c_{a b}\,A_{a \alpha}\,A_{b \beta},~~\alpha, \beta=1,2,\ldots, M.\end{aligned}$$

[**(b)**]{} Let $v_{a}=A_{a \alpha}\,\bar{v}_{\alpha}$ where a bar over a letter denotes complex conjugation. Then (\[coupledmKdVN\]) consistently reduce to $$\label{eqn2}
\mu v_{\alpha,t}+3 \rho \,v_{\alpha,x}+v_{\alpha,xxx}=0,~~\alpha=1,2,\ldots, M,$$ where $$\rho=c_{\alpha \beta} v_{\alpha} v_{\beta}+2c_{\alpha a}\, A_{a \beta} v_{\alpha} \bar{v}_{\beta}+c_{ab} A_{a \alpha} A_{b \beta} \bar{v}_{\alpha}\, \bar{v}_{\beta}$$ with the following constraints $$\begin{aligned}
\bar{c}_{\alpha \beta}&=&c_{ab}\, A_{a \alpha}\, 