the parameters $k_1, k_2$ are real then the Example 1 given in part i. is also valid for this case.

Nonlocal Reductions for N=4
---------------------------

Here we present two different nonlocal reductions of the coupled HI-mKdV system (\[coupledmKdVN\]) for $N=4$.\
**i.** $v_4=a_1v_2^{\varepsilon}$, $v_3=b_1v_1^{\varepsilon}$, $v_i^{\varepsilon}=v_i(\varepsilon_1 t,\varepsilon_2 x)$, $\varepsilon_i^2=1$, $i=1, 2$, $a_1$ and $b_1$ are constants. We have a consistent reduction if the following conditions hold: $$\varepsilon_1\varepsilon_2=1, \quad \rho=\rho^{\varepsilon}.$$ To have a nonlocal system we have only one choice $(\varepsilon_1,\varepsilon_2)=(-1,-1)$. The relation $\rho=\rho^{\varepsilon}$ is satisfied if $$\label{N=4nonlocalcondi}
1)\, c_{12}=c_{34}a_1b_1,\quad 2)\, c_{14}a_1=c_{23}b_1.$$ Then the reduced system is $$\begin{aligned}
&\mu v_{1,t}(t,x)+6[c_{12}v_1(t,x)v_{2}(t,x)+c_{13}b_1v_1(t,x)v_1(-t,-x)+c_{23}b_1v_1(t,x)v_2(-t,-x)\nonumber\\
&+c_{23}b_1v_2(t,x)v_1(-t,-x)+c_{24}a_1v_2(t,x)v_2(-t,-x)+c_{12}v_1(-t,-x)v_2(-t,-x)]v_{1,x}(t,x)\nonumber\\
&+v_{1,xxx}(t,x)=0,\nonumber\\
 &\mu v_{2,t}(t,x)+6[c_{12}v_1(t,x)v_{2}(t,x)+c_{13}b_1v_1(t,x)v_1(-t,-x)+c_{23}b_1v_1(t,x)v_2(-t,-x)\nonumber\\
&+c_{23}b_1v_2(t,x)v_1(-t,-x)+c_{24}a_1v_2(t,x)v_2(-t,-x)+c_{12}v_1(-t,-x)v_2(-t,-x)]v_{2,x}(t,x)\nonumber\\
&+v_{2,xxx}(t,x)=0.\label{nonlocalreducediN=4}\end{aligned}$$

To obtain one-soliton solution of the system (\[nonlocalreducediN=4\]), we use the Type 2 approach since Type 1 gives trivial solution. Hence we take $$\begin{aligned}
&g_4f^{\varepsilon}-a_1g_2^{\varepsilon}=0,\label{relationnonlocalia}\\
&g_3f^{\varepsilon}-b_1g_1^{\varepsilon}=0,\label{relationnonlocalib}\end{aligned}$$ to get constraints for the parameters of one-soliton solution of the system (\[nonlocalreducediN=4\]). Here we obtain the following cases satisfying both (\[relationnonlocalia\]) and (\[relationnonlocalib\]).\
**Case A.** One of the sets of the relations solving the equation (\[relationnonlocalia\]) is $$\label{caseA}
1)\, c_{12}=c_{13}=c_{23}=0, \quad 2)\, e^{\delta_2}=\sigma_1\frac{(k_2+k_4)}{\sqrt{a_1c_{24}}},\quad 3)\, e^{\delta_4}=\sigma_2\sqrt{a_1}\frac{(k_2+k_4)}{\sqrt{c_{24}}}, \sigma_i=\pm 1, i=1, 2.$$ In this case the system (\[nonlocalreducediN=4\]) becomes $$\begin{aligned}
&\mu v_{1,t}(t,x)+6c_{24}a_1v_2(t,x)v_2(-t,-x)v_{1,x}(t,x)+v_{1,xxx}(t,x)=0,\nonumber\\
 &\mu v_{2,t}(t,x)+6c_{24}a_1v_2(t,x)v_2(-t,-x)v_{2,x}(t,x)+v_{2,xxx}(t,x)=0.\label{nonlocalreducediAN=4}\end{aligned}$$

From the equation (\[relationnonlocalib\]) we get the following cases:\
