x)+c_{23}a_1b_1]v_{1,x}(t,x)+v_{1,xxx}(t,x)=0.\end{aligned}$$

Therefore we have three different nonlocal cmKdV equations:\
**a.** **T-Symmetric Nonlocal CMKdV Equation:** $$\begin{aligned}
&\mu v_{1,t}(t,x)+6[\bar{c}_{23}\bar{a}_3\bar{b}_3v_1^2(t,x)+(c_{12}b_1+c_{13}a_1+c_{23}a_2b_1+c_{23}a_1b_2)v_1(t,x)\nonumber\\
&+c_{23}a_3b_3\bar{v}_1^2(-t,x)+(c_{23}a_1b_3+c_{23}a_3b_1)\bar{v}_1(-t,x)\nonumber\\
&+(c_{12}b_3+c_{13}a_3+c_{23}a_2b_3+c_{23}a_3b_2)v_1(t,x)\bar{v}_1(-t,x)+c_{23}a_1b_1]v_{1,x}(t,x)+v_{1,xxx}(t,x)=0,\end{aligned}$$ where $\mu=-\bar{\mu}$ and the conditions (\[condnonlocalivN=3\]) hold.\
**b.** **S-Symmetric Nonlocal CMKdV Equation:** $$\begin{aligned}
&\mu v_{1,t}(t,x)+6[\bar{c}_{23}\bar{a}_3\bar{b}_3v_1^2(t,x)+(c_{12}b_1+c_{13}a_1+c_{23}a_2b_1+c_{23}a_1b_2)v_1(t,x)\nonumber\\
&+c_{23}a_3b_3\bar{v}_1^2(t,-x)+(c_{23}a_1b_3+c_{23}a_3b_1)\bar{v}_1(t,-x)\nonumber\\
&+(c_{12}b_3+c_{13}a_3+c_{23}a_2b_3
+c_{23}a_3b_2)v_1(t,x)\bar{v}_1(t,-x)+c_{23}a_1b_1]v_{1,x}(t,x)+v_{1,xxx}(t,x)=0,\end{aligned}$$ where $\mu=-\bar{\mu}$ and the conditions (\[condnonlocalivN=3\]) hold.\
**c.** **ST-Symmetric Nonlocal CMKdV Equation:** $$\begin{aligned}
&\mu v_{1,t}(t,x)+6[\bar{c}_{23}\bar{a}_3\bar{b}_3v_1^2(t,x)+(c_{12}b_1+c_{13}a_1+c_{23}a_2b_1+c_{23}a_1b_2)v_1(t,x)\nonumber\\
&+c_{23}a_3b_3\bar{v}_1^2(-t,-x)+(c_{23}a_1b_3+c_{23}a_3b_1)\bar{v}_1(-t,-x)\nonumber\\
&+(c_{12}b_3+c_{13}a_3+c_{23}a_2b_3+c_{23}a_3b_2)v_1(t,x)\bar{v}_1(-t,-x)+c_{23}a_1b_1]v_{1,x}(t,x)+v_{1,xxx}(t,x)=0,\end{aligned}$$ where $\mu=\bar{\mu}$ and the conditions (\[condnonlocalivN=3\]) hold.\
By using Type 1 approach, we obtain the constraints that the parameters of one-soliton solution of the equation (\[nonlocalivN=3\]) as $$\begin{aligned}
&1)\, a_1=b_1=0,\quad 2)\, k_1=k_2=k_3=\varepsilon_2\bar{k}_1,\quad  3)\, e^{\delta_2}=b_2e^{\delta_1}+b_3e^{\bar{\delta}_1},\nonumber\\
&4)\, e^{\delta_3}=a_2e^{\delta_1}+a_3e^{\bar{\delta}_1},\quad  5)\, c_{ij}e^{\delta_i+\delta_j}=\bar{c}_{ij}e^{\bar{\delta}_i+\bar{\delta}_j},\, i, j=1, 2, 3.\end{aligned}$$ With these conditions, the equation (\[nonlocalivN=3\]) becomes the following nonlocal cmKdV equation: $$\begin{aligned}
\label{reducednonlocalivN=3}
&\mu v_{1,t}(t,x)+6[(c_{12}b_2+c_{13}a_2+c_{23}a_2b_2)v_1^2(t,x)+c_{23}a_3b_3\bar{v}_1^2(\varepsilon_1t,\varepsilon_2 x)\nonumber\\
&+(c_{12}b_3+c_{13}a_3+c_{23}a_2b_3+c_{23}a_3b_2)v_1(t,x)\bar{v}_1(\varepsilon_1t,\varepsilon_2 x)]v_{1,x}(t,x)+v_{1,xxx}(t,x)=0.\end{aligned}$$ One-soliton solution of the equation (\[reducednonlocalivN=3\]) is $$v_1(t,x)=e^{k_1x-\frac{k_1^3}{\mu}t+\delta_1}.$$

For $N=3$ we have reductions to system of two equations, but if we require Type 1 one-soliton solution, they reduce to a single equation.

Acknowledgment
==============

This work is partially supported by the Scientific and Technological Research Council of Turkey (TÜBİTAK).

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