system (\[nonlocaliN=3\]) are obtained from $$\frac{g_3}{f}=a_2\frac{g_2^{\epsilon}}{f^{\epsilon}}+a_1\frac{g_1^{\epsilon}}{f^{\epsilon}} \Rightarrow f=f^{\epsilon},\, g_3=a_2g_2^{\epsilon}+a_1g_1^{\epsilon}$$ as $$1)\, k_1=k_2=-k_3,\quad 2)\, e^{\delta_3}=a_2e^{\delta_2}+a_1e^{\delta_1},\quad 3)\, c_{12}=c_{13}=c_{23}=0.$$ The above conditions yields that $v_2=e^{\delta_2-\delta_1}v_1$. The nonlocal system (\[nonlocaliN=3\]) reduces to local linear equation $$\mu v_{1,t}(t,x)+v_{1,xxx}(t,x)=0,$$ and one-soliton solution of the above equation is $$v_1=e^{k_1x-\frac{k_1^3}{\mu}t+\delta_1}.$$

**ii.** $v_3(t,x)=a_3+a_2\bar{v}_2(\varepsilon_1t,\varepsilon_2 x)+a_1\bar{v}_1(\varepsilon_1t,\varepsilon_2 x)=a_3+a_2\bar{v}_2^{\varepsilon}+a_1\bar{v}_1^{\varepsilon}$, $\varepsilon_j^2=1$, $j=1, 2$, and $a_i$, $i=1, 2, 3$ are constants. Applying this reduction to (\[N=3c\]) gives the constraints on the parameters as $$\label{condnonlocaliiN=3}
1)\, \mu=\bar{\mu}\varepsilon_1\varepsilon_2,\, 2)\, a_3=0,\, 3)\, c_{12}=0,\, 4)\, c_{13}a_2=\bar{c}_{23}\bar{a}_1,\, 5)\, c_{13}a_1=\bar{c}_{13}\bar{a}_1,\,
6)\, c_{23}a_2=\bar{c}_{23}\bar{a}_2$$ for consistent reduction. Hence we have $$\begin{aligned}
&\mu v_{1,t}(t,x)+6[c_{13}v_1(t,x)+c_{23}v_2(t,x)][a_1\bar{v}_1(\epsilon_1t,\epsilon_2x)+a_2\bar{v}_2(\epsilon_1t,\epsilon_2x)]v_{1,x}(t,x)
+v_{1,xxx}(t,x)=0,\nonumber\\
&\mu v_{2,t}(t,x)+6[c_{13}v_1(t,x)+c_{23}v_2(t,x)][a_1\bar{v}_1(\epsilon_1t,\epsilon_2x)+a_2\bar{v}_2(\epsilon_1t,\epsilon_2x)]v_{2,x}(t,x)
+v_{2,xxx}(t,x)=0.\label{NONLOCALiiN=3}\end{aligned}$$ According to $(\epsilon_1,\epsilon_2)=\{(-1,1),(1,-1),(-1,-1)\}$ we have three different nonlocal cmKdV systems:\
**a.** **T-Symmetric Nonlocal CMKdV System:** $$\begin{aligned}
&\mu v_{1,t}(t,x)+6[c_{13}v_1(t,x)+c_{23}v_2(t,x)][a_1\bar{v}_1(-t,x)+a_2\bar{v}_2(-t,x)]v_{1,x}(t,x)+v_{1,xxx}(t,x)=0,\nonumber\\
&\mu v_{2,t}(t,x)+6[c_{13}v_1(t,x)+c_{23}v_2(t,x)][a_1\bar{v}_1(-t,x)+a_2\bar{v}_2(-t,x)]v_{2,x}(t,x)+v_{2,xxx}(t,x)=0,\label{TsymmN=3}\end{aligned}$$ where $\mu=-\bar{\mu}$ and the conditions (\[condnonlocaliiN=3\]) hold.\
**b.** **S-Symmetric Nonlocal CMKdV System:** $$\begin{aligned}
&\mu v_{1,t}(t,x)+6[c_{13}v_1(t,x)+c_{23}v_2(t,x)][a_1\bar{v}_1(t,-x)+a_2\bar{v}_2(t,-x)]v_{1,x}(t,x)+v_{1,xxx}(t,x)=0,\nonumber\\
&\mu v_{2,t}(t,x)+6[c_{13}v_1(t,x)+c_{23}v_2(t,x)][a_1\bar{v}_1(t,-x)+a_2\bar{v}_2(t,-x)]v_{2,x}(t,x)+v_{2,xxx}(t,x)=0,\label{SsymmN=3}\end{aligned}$$ where $\mu=-\bar{\mu}$ and the conditions (\[condnonlocaliiN=3\]) hold.\
**c.** **ST-Symmetric Nonlocal CMKdV System:** $$\begin{aligned}
&\mu v_{1,t}(t,x)+6[c_{13}v_1(t,x)+c_{23}v_2(t,x)][a_1\bar{v}_1(-t,-x)+a_2\bar{v}_2(-t,-x)]v_{1,x}(t,x)+v_{1,xxx}(t,x)=0,\nonumber\\
&\mu v_{2,t}(t,x)+6[c_{13}v_1(t,x)+c_{23}v_2(t,x)][a_1\bar{v}_1(-t,-x)+a_2\bar{v}_2(-t,-x)]v_{2,x}(t,x)+v_{2,xxx}(t,x)=0,\label{STsymmN=3}\end{aligned}$$ where $\mu=\bar{\mu}$ and the conditions (\[condnonlocaliiN=3\]) hold.\
When we apply the Type 1 approach, the constraints that the parameters of the one-soliton solution of the system (\[NONLOCALiiN=3\]) are obtained as $$1)\, k_3=\epsilon_2\bar{k}_2=\epsilon_2\bar{k}_1,\quad 2)\, e^{\delta_3}=a_2e^{\bar{\delta}_2}+a_1e^{\bar{\delta}_1},
\quad 3)\, c_{13}e^{\delta_1+\delta_3}=\bar{c_{13}}e^{\bar{\delta}_1+\bar{\delta}_3}, \quad 4)\, c_{23}e^{\delta_2+\delta_3}=\bar{c_{23}}e^{\bar{\delta}_2+\bar{\delta}_3}.$$ In this case $v_2(t,x)=e^{\delta_2-\delta_1}v_1(t,x)$ and the system (\[NONLOCALiiN=3\]) reduces to a single 