and Type 2 approaches [@GurPek2].//

**i.** $v_2=a_2+a_1v_1$, $a_i$, $i=1, 2$ are constants. When we apply the reduction to the equation (\[N=2b\]) we obtain the equation (\[N=2a\]) $$\mu v_{1,t}+3\rho v_{1,x}+v_{1,xxx}=0,$$ without any additional condition on the parameters. Hence the reduction is consistent and the reduced equation is $$\label{localiN=2}
\mu v_{1,t}+6\alpha v_1v_{1,x}+6\beta v_1^2v_{1,x}+v_{1,xxx}=0$$ for $\alpha=c_{12}a_2$ and $\beta=c_{12}a_1$. This equation is a combination of KdV and mKdV equations.

If we follow the Type 1 approach, the constraints that the parameters of the one-soliton solution of (\[localiN=2\]) are obtained from $$\frac{g_2}{f}=a_2+a_1\frac{g_1}{f} \Rightarrow g_2=a_2f+a_1g_1,$$ as $$1)\, a_2=0,\quad 2)\, k_2=k_1,\quad 3)\, e^{\delta_2}=a_1e^{\delta_1}.$$

**ii.** $v_2=a_2+a_1\bar{v}_1$, $a_i$, $i=1, 2$ are constants. We use this reduction in (\[N=2b\]) and obtain the following conditions to have consistent reduction: $$\label{condlocaliiN=2}
1)\,\,  \mu=\bar{\mu},\,\, 2)\,\,  a_2=0, \,\, 3)\,\,  c_{12}a_1=\bar{c}_{12}\bar{a}_1.$$ So the reduced equation is the complex mKdV (cmKdV) equation $$\label{localiiN=2}
\mu v_{1,t}+6\alpha |v_1|^2v_{1,x}+v_{1,xxx}=0,$$ where $\alpha=c_{12}a_1$ and $\mu$ are real numbers.

From the Type 1 approach, the constraints that the parameters of the one-soliton solution of (\[localiiN=2\]) are obtained from $$\frac{g_2}{f}=a_1\frac{\bar{g}_1}{\bar{f}} \Rightarrow g_2=a_1\bar{g}_1,\quad f=\bar{f}$$ as $$1)\, k_2=\bar{k}_1,\quad 2)\, e^{\delta_2}=a_1e^{\bar{\delta}_1},$$ besides the conditions (\[condlocaliiN=2\]).

Nonlocal Reductions for N=2
---------------------------

**i.** **$v_2(t,x)=a_2+a_1v_1(\varepsilon_1t,\varepsilon_2 x)=a_2+a_1v_1^{\varepsilon}$**, $\varepsilon_i^2=1$, $a_i$, $i=1, 2$ are constants. When we use this reduction in (\[N=2b\]) 