for consistency of reduction we get the following conditions: $$1)\,\, \varepsilon_1\varepsilon_2=1,\,\, 2)\,\, a_2=0.$$ Therefore to have a nonlocal equation, there is only one possibility; $(\varepsilon_1,\varepsilon_2)=(-1,-1)$. The reduced equation is ST-symmetric nonlocal mKdV equation, $$\label{nonlocaliN=2}
\mu v_{1,t}(t,x)+6\alpha v_1(t,x)v_1(-t,-x)v_{1,x}(t,x)+v_{1,xxx}(t,x)=0,$$ where $\alpha=c_{12}a_1$.

In this case, if we use the Type 1 approach to find one-soliton solution of (\[nonlocaliN=2\]) we get trivial solution. Hence we use the Type 2 approach. Therefore the constraints that the parameters of the one-soliton solution of (\[localiiN=2\]) are obtained from $$g_2f^{\varepsilon}-a_1fg_1^{\varepsilon}=0,\, \mathrm{where}\,  g_1^{\varepsilon}=g_1(-t,-x),\, f^{\varepsilon}=f(-t,-x),$$ as $$1)\, e^{\delta_1}=\pm\frac{(k_1+k_2)}{\sqrt{c_{12}a_1}} ,\quad 2)\, e^{\delta_2}=\pm\frac{\sqrt{a_1}(k_1+k_2)}{\sqrt{c_{12}}}.$$

**ii.** **$v_2(t,x)=a_2+a_1\bar{v}_1(\varepsilon_1t,\varepsilon_2 x)=a_2+a_1\bar{v}_1^{\varepsilon}$**, $\varepsilon_i^2=1$, $a_i$, $i=1, 2$ are constants. Applying this reduction to (\[N=2b\]) gives the constraints on the parameters as $$\label{condnonlocaliiN=2}
1)\, \mu=\bar{\mu}\varepsilon_1\varepsilon_2,\quad 2)\, a_2=0,\quad 3)\, c_{12}a_1=\bar{c}_{12}\bar{a}_1.$$ Hence we have three different nonlocal cmKdV equations:\
**a.** **T-Symmetric Nonlocal CMKdV Equation:** $$\label{TsymmN=2}
\mu v_{1,t}(t,x)+6\alpha v_1(t,x)\bar{v}_1(-t,x)+v_{1,xxx}(t,x)=0,$$ where $\mu=-\bar{\mu}$ and $\alpha=c_{12}a_1\in \mathbb{R}$.\
**b.** **S-Symmetric Nonlocal CMKdV Equation:** $$\label{SsymmN=2}
\mu v_{1,t}(t,x)+6\alpha v_1(t,x)\bar{v}_1(t,-x)+v_{1,xxx}(t,x)=0,$$ where $\mu=-\bar{\mu}$ and $\alpha=c_{12}a_1\in \mathbb{R}$.\
**c.** **ST-Symmetric Nonlocal CMKdV Equation:** $$\label{STsymmN=2}
\mu v_{1,t}(t,x)+6\alpha v_1(t,x)\bar{v}_1(-t,-x)+v_{1,xxx}(t,x)=0,$$ where $\mu=\bar{\mu}$ and $\alpha=c_{12}a_1\in \mathbb{R}$.\
If we use the Type 1 approach, we get the constraints that the parameters of the one-soliton solutions of (\[TsymmN=2\])-(\[STsymmN=2\]) from $\frac{g_2}{f}=a_1\frac{\bar{g}_1^{\varepsilon}}{\bar{f}^{\varepsilon}}$ that is $$g_2=a_1\bar{g}_1^{\varepsilon},\, f=\bar{f}^{\varepsilon},\, \mathrm{where}\, \bar{g}_1^{\varepsilon}=\bar{g}_1(\varepsilon_1 t,\varepsilon_2 x),\,
\bar{f}^{\varepsilon}=\bar{f}(\varepsilon_1 