nonlocal cmKdV equation [@GurPek2] $$\mu v_1(t,x)+6(c_{13}+c_{23}e^{\delta_2-\delta_1})(a_1+a_2e^{\bar{\delta}_2-\bar{\delta}_1})v_1(t,x)
\bar{v}_1(\epsilon_1t,\epsilon_2x)v_{1,x}(t,x)+v_{1,xxx}(t,x)=0.$$ One-soliton solution of the above equation is $$v_1=\frac{e^{k_1x-\frac{k_1^3}{\mu}t+\delta_1}}{1+\frac{1}{(k_1+\epsilon_2\bar{k}_1)^2}[c_{13}e^{\delta_1}+c_{23}e^{\delta_2}]
[a_2e^{\bar{\delta}_2}+a_1e^{\bar{\delta}_1}]e^{(k_1+\epsilon_2\bar{k}_1)x-\frac{(k_1^3+\epsilon_2\bar{k}_1^3)}{\mu}t}},$$ with the conditions (\[condnonlocaliiN=3\]) hold.

**iii.** $v_3(t,x)=a_1+a_2v_1(t,x)+a_3v_1(\varepsilon_1t,\varepsilon_2 x)=a_1+a_2v_1+a_3v_1^{\varepsilon}$, $v_2(t,x)=b_1+b_2v_1(t,x)+b_3v_1(\varepsilon_1t,\varepsilon_2 x)=b_1+b_2v_1+b_3v_1^{\varepsilon}$ where $a_i, b_i$, $i=1, 2, 3$ are constants, and $\varepsilon_j^2=1$, $j=1, 2$. Applying this reduction to (\[N=3b\]) and (\[N=3c\]) gives the constraints on the parameters as $$\begin{aligned}
\label{condnonlocaliiiN=3}
&1)\, \varepsilon_1\varepsilon_2=1,\nonumber\\
&2)\, c_{12}b_1+c_{13}a_1+c_{23}a_2b_1+c_{23}a_1b_2=c_{23}a_1b_3+c_{23}a_3b_1,\nonumber\\
&3)\, c_{12}b_2+c_{13}a_2+c_{23}a_2b_2=c_{23}a_3b_3,\end{aligned}$$ for consistent reduction. Here we have one possibility; $(\varepsilon_1,\varepsilon_2)=(-1,-1)$. The reduced equation is $$\begin{aligned}
\label{nonlocaliiiN=3}
&\mu v_{1,t}(t,x)+6[(c_{12}b_2+c_{13}a_2+c_{23}a_2b_2)v_1^2(t,x)+(c_{12}b_1+c_{13}a_1+c_{23}a_2b_1+c_{23}a_1b_2)v_1(t,x)\nonumber\\
&+c_{23}a_3b_3v_1^2(-t,-x)+(c_{23}a_1b_3+c_{23}a_3b_1)v_1(-t,-x)\nonumber\\
&+(c_{12}b_3+c_{13}a_3+c_{23}a_2b_3+c_{23}a_3b_2)v_1(t,x)v_1(-t,-x)+c_{23}a_1b_1]v_{1,x}(t,x)+v_{1,xxx}(t,x)=0,\end{aligned}$$ with the conditions (\[condnonlocaliiiN=3\]) hold. If we use the Type 1 approach to obtain the one-soliton solution of the equation we get the following constraints: $$1)\, a_1=a_3=b_1=b_2=0,\, 2)\, k_1=-k_2=k_3,\, 3)\, c_{12}=c_{13}=c_{23}=0,\, 4)\, e^{\delta_3}=a_2e^{\delta_1},\, 5)\, e^{\delta_2}=b_3e^{\delta_1}.$$ Hence the equation (\[nonlocaliiiN=3\]) reduces to the local linear equation $$\mu v_{1,t}(t,x)+v_{1,xxx}(t,x)=0,$$ and one-soliton solution of this equation is $$v_1=e^{k_1x-\frac{k_1^3}{\mu}t+\delta_1}.$$

**iv.** $v_3(t,x)=a_1+a_2v_1(t,x)+a_3\bar{v}_1(\varepsilon_1t,\varepsilon_2 x)=a_1+a_2v_1+a_3\bar{v}_1^{\varepsilon}$, $v_2(t,x)=b_1+b_2v_1(t,x)+b_3\bar{v}_1(\varepsilon_1t,\varepsilon_2 x)=b_1+b_2v_1+b_3\bar{v}_1^{\varepsilon}$ where $a_i, b_i$, $i=1, 2, 3$ are constants, and $\varepsilon_j^2=1$, $j=1, 2$. When we apply this reduction to (\[N=3b\]) and (\[N=3c\]) we obtain the constraints on the parameters as $$\begin{aligned}
\label{condnonlocalivN=3}
&1)\, \bar{\mu}\varepsilon_1\varepsilon_2=\mu,\nonumber\\
&2)\, c_{12}b_1+c_{13}a_1+c_{23}a_2b_1+c_{23}a_1b_2=\bar{c}_{23}\bar{a}_1\bar{b}_3+\bar{c}_{23}\bar{a}_3\bar{b}_1,\nonumber\\
&3)\, c_{12}b_2+c_{13}a_2+c_{23}a_2b_2=\bar{c}_{23}\bar{a}_3\bar{b}_3,\nonumber\\
&4)\, c_{12}b_3+c_{13}a_3+c_{23}a_3b_2+c_{23}a_2b_3=\bar{c}_{12}\bar{b}_3+\bar{c}_{13}\bar{a}_3+
\bar{c}_{23}\bar{a}_3\bar{b}_2+\bar{c}_{23}\bar{a}_2\bar{b}_3,\nonumber\\
&5\, c_{23}a_1b_1=\bar{c}_{23}\bar{a}_1\bar{b}_1,\end{aligned}$$ for consistent reduction. Hence we have a single nonlocal cmKdV equation $$\begin{aligned}
\label{nonlocalivN=3}
&\mu v_{1,t}(t,x)+6[\bar{c}_{23}\bar{a}_3\bar{b}_3v_1^2(t,x)+(c_{12}b_1+c_{13}a_1+c_{23}a_2b_1+c_{23}a_1b_2)v_1(t,x)\nonumber\\
&+c_{23}a_3b_3\bar{v}_1^2(\varepsilon_1t,\varepsilon_2 x)+(c_{23}a_1b_3+c_{23}a_3b_1)\bar{v}_1(\varepsilon_1t,\varepsilon_2 x)\nonumber\\
&+(c_{12}b_3+c_{13}a_3+c_{23}a_2b_3+c_{23}a_3b_2)v_1(t,x)\bar{v}_1(\varepsilon_1t,\varepsilon_2 