**Case A.I.** In addition to the constraints (\[caseA\]), one of the sets of the conditions that the equation (\[relationnonlocalib\]) yields, is $\{ k_1=k_2, k_3=k_4, e^{\delta_3}=\sigma_1\sigma_2b_1e^{\delta_1}\}$ for $\sigma_i=\pm 1, i=1, 2.$. Hence one-soliton solution of the system (\[nonlocalreducediAN=4\]) is $$\displaystyle
v_1=\frac{e^{k_2x-\frac{k_2^3}{a}t+\delta_1}}{1+\sigma_1\sigma_2e^{(k_2+k_4)x-\frac{(k_2^3+k_4^3)}{a}t}},\,\,
v_2=\frac{\sigma_1(k_2+k_4)e^{k_2x-\frac{k_2^3}{a}t}}{\sqrt{a_1c_{24}}[1+\sigma_1\sigma_2e^{(k_2+k_4)x-\frac{(k_2^3+k_4^3)}{a}t}]}.$$ Clearly, here we have $v_2=\xi v_1$, where $\xi=\frac{\sigma_1(k_2+k_4)}{e^{\delta_1}\sqrt{a_1c_{24}}}$. Hence the system (\[nonlocalreducediAN=4\]) reduces to a single nonlocal ST-symmetric mKdV equation [@GurPek2] $$\mu v_{1,t}(t,x)+6c_{24}a_1\xi^2v_1(t,x)v_1(-t,-x)v_{1,x}(t,x)+v_{1,xxx}(t,x)=0.$$ Note that another set of the conditions $\{ k_1=k_4, k_2=k_3, e^{\delta_3}=\sigma_1\sigma_2b_1e^{\delta_1}\}$ yielded form (\[relationnonlocalib\]) gives similar solution.

**Example 2.** If we take the parameters as $k_4=2, \mu=k_2=e^{\delta_1}=a_1=c_{24}=\sigma_1=\sigma_2=1$, then one-soliton solution becomes $$\displaystyle
v_1=\frac{e^{x-t}}{1+e^{3x-9t}},$$ and $v_2=3v_1$. This solution is finite and bounded for any $(x,t)$. The graph of the solution $v_1$ is given in Figure 2.

![One-soliton solution for (\[nonlocalreducediAN=4\]) with the parameters $k_4=2, \mu=k_2=e^{\delta_1}=a_1=c_{24}=\sigma_1=\sigma_2=1$.](figurec.jpg)

**Case A.II.** In addition to the constraints (\[caseA\]), another set of the conditions obtained from the equation (\[relationnonlocalib\]) is $\displaystyle \{ k_1=-k_3, k_2=-\frac{k_3^2}{k_4}, e^{\delta_3}=-b_1e^{\delta_1}\}$. In this case one-soliton solution of the system (\[nonlocalreducediAN=4\]) is $$\begin{aligned}
\displaystyle
&v_1=\frac{e^{-k_3x+\frac{k_3^3}{\mu}t+\delta_1}-\sigma_1\sigma_2e^{(k_4-k_3-\frac{k_3^2}{k_4})x-\frac{1}{\mu}(k_4^3-k_3^3-\frac{k_3^6}{k_4^3})t+\delta_1}   }{1+\sigma_1\sigma_2e^{(k_4-\frac{k_3^2}{k_4})x-\frac{(k_4^3-\frac{k_3^6}{k_4^3})}{\mu}t}},\\
&v_2=\frac{\sigma_1(k_4-\frac{k_3^2}{k_4})e^{-\frac{k_3^2}{k_4}x+\frac{k_3^6}{\mu k_4^3}t}   }{\sqrt{a_1c_{24}}[1+\sigma_1\sigma_2e^{(k_4-\frac{k_3^2}{k_4})x-\frac{(k_4^3-\frac{k_3^6}{k_4^3})}{\mu}t}]},\end{aligned}$$ for $\sigma_i=\pm 1, i=1, 2$.\
**Example 3.** Take the parameters as $k_4=4, \mu=10, k_3=e^{\delta_1}=a_1=c_{24}=\sigma_1=\sigma_2=1$. Therefore one-soliton solution of (\[nonlocalreducediAN=4\]) becomes $$\displaystyle
v_1=\frac{e^{-x+\frac{1}{10}t}-e^{\frac{11}{4}x-\frac{4031}{640}t}}{1+e^{\frac{15}{4}x-\frac{819}{128}t}},
\quad v_2=\frac{15e^{-\frac{1}{4}x+\frac{1}{640}t}}{4(1+e^{\frac{15}{4}x-\frac{819}{128}t})}.$$ The solutions are finite but not bounded. 