system (\[nonlocalreducediBN=4\]) is given by $$\displaystyle
v_1=\frac{g_1}{f}, \quad v_2=\frac{g_2}{f},$$ where $$\begin{aligned}
g_1&=e^{k_1x-\frac{k_1^3}{\mu}t+\delta_1}+\frac{(k_1-k_4)c_{24}}{(k_1+k_4)k_4^2}e^{(k_1+k_4)x-\frac{(k_1^3+k_4^3)}{\mu}t+\delta_1+\delta_2+\delta_4},\\
g_2&=e^{\delta_2}+\frac{c_{13}}{(k_1+k_3)^2}e^{(k_1+k_3)x-\frac{(k_1^3+k_3^3)}{\mu}t+\delta_1+\delta_2+\delta_3},\\
f&=1+\frac{c_{13}}{(k_1+k_3)^2}e^{(k_1+k_3)x-\frac{(k_1^3+k_3^3)}{\mu}t+\delta_1+\delta_3}
+\frac{c_{24}}{k_4^2}e^{k_4x-\frac{k_4^3}{\mu}t+\delta_2+\delta_4}\nonumber\\
&+\frac{(k_1-k_4)(k_3-k_4)c_{13}c_{24}}{(k_1+k_4)(k_3+k_4)(k_1+k_3)^2k_4^2}e^{(k_1+k_3+k_4)x-\frac{(k_1^3+k_3^3+k_4^3)}{\mu}t+\delta_1+\delta_2+\delta_3+\delta_4}.\end{aligned}$$

**Example 5.** Let us take the parameters as $k_3=2, a=4, k_4=a_1=b_1=c_{13}=c_{24}=\sigma_1=\sigma_2=1, e^{\delta_4}=2$. In this case one-soliton solution of (\[nonlocalreducediBN=4\]) becomes $$\displaystyle
v_1=\frac{-3\sqrt{3}[2e^{7x-\frac{343}{4}t}+3e^{8x-86t}]}{1+2e^{9x-\frac{351}{4}t}+2e^{x-\frac{1}{4}t}+e^{10x-88t}},\quad v_2=\frac{1+2e^{9x-\frac{351}{4}t}}{1+2e^{9x-\frac{351}{4}t}+2e^{x-\frac{1}{4}t}+e^{10x-88t}}.$$ These solutions are finite and bounded. The graphs of the functions $v_1$ and $v_2$ are given in Figure 5(a) and Figure 5(b) respectively.

**Case C.** The last set of the constraints solving both of the equations (\[relationnonlocalia\]) and (\[relationnonlocalib\]) is $$\begin{aligned}
\label{caseC}\displaystyle
&1)\,c_{12}=c_{23}=0,\quad 2)\, e^{\delta_1}=\sigma_1(k_1+k_3)\sqrt{\frac{(k_1+k_2)(k_1+k_4)}{(k_1-k_2)(k_4-k_1)b_1c_{13}}},\nonumber\\
&3)\, e^{\delta_2}=\sigma_2(k_2+k_4)\sqrt{\frac{(k_1+k_2)(k_2+k_3)}{(k_1-k_2)(k_2-k_3)a_1c_{24}}},\quad
4)\, e^{\delta_3}=\sigma_3(k_1+k_3)\sqrt{b_1\frac{(k_2+k_3)(k_3+k_4)}{(k_2-k_3)(k_4-k_3)c_{13}}},\nonumber\\
&5)\, e^{\delta_4}=\sigma_4(k_2+k_4)\sqrt{a_1\frac{(k_1+k_4)(k_3+k_4)}{(k_1-k_4)(k_3-k_4)c_{24}}},\, \sigma_i=\pm 1, i=1,2,3,4.\end{aligned}$$ Here the reduced system is the same with (\[nonlocalreducediBN=4\]). One-soliton solution of this system is given by $$\displaystyle
v_1=\frac{g_1}{f}, \quad v_2=\frac{g_2}{f},$$ where $$\begin{aligned}
g_1&=e^{k_1x-\frac{k_1^3}{\mu}t+\delta_1}+\frac{(k_1-k_2)(k_1-k_4)c_{24})}{(k_1+k_2)(k_1+k_4)(k_2+k_4)^2}e^{(k_1+k_2+k_4)x
-\frac{(k_1^3+k_2^3+k_4^3)}{\mu}t+\delta_1+\delta_2+\delta_4},\\
g_2&=e^{k_2x-\frac{k_2^3}{\mu}t+\delta_2}-\frac{(k_1-k_2)(k_2-k_3)c_{13}}{(k_1+k_2)(k_2+k_3)(k_1+k_3)^2}e^{(k_1+k_2+k_3)x-\frac{(k_1^3+k_2^3+k_3^3)}{\mu}t
+\delta_1+\delta_2+\delta_3},\\
f&=1+\frac{c_{13}}{(k_1+k_3)^2}e^{(k_1+k_3)x-\frac{(k_1^3+k_3^3)}{\mu}t+\delta_1+\delta_3}
+\frac{c_{24}}{(k_2+k_4)^2}e^{(k_2+k_4)x-\frac{(k_2^3+k_4^3)}{\mu}t+\delta_2+\delta_4}\nonumber\\
&-\frac{(k_1-k_2)(k_1-k_4)(k_2-k_3)(k_3-k_4)c_{13}c_{24}e^{(k_1+k_2+k_3+k_4)x-\frac{(k_1^3+k_2^3+k_3^3+k_4^3)}
{\mu}t+\delta_1+\delta_2+\delta_3+\delta_4}}{(k_1+k_2)(k_1+k_4)(k_2+k_3)(k_3+k_4)(k_1+k_3)^2(k_2+k_4)^2}.\end{aligned}$$

**Example 6.** Let us choose the parameters as $k_1=4, k_2=1, k_3=3, k_4=2, \mu=3, a_1=b_1=c_{13}=c_{24}=\sigma_i=1, i=1, 2, 3, 4$. Hence we obtain the following one-soliton solution of (\[nonlocalreducediBN=4\]): $$\displaystyle
|v_1|^2=\frac{W_1}{Y}, \quad |v_2|^2=\frac{W_2}{Y^2},$$ where $$\begin{aligned}
W_1&=245(2e^{6x}+e^{6t})e^{8x+18t},\\
W_2&=15e^{2x+\frac{14}{3}t}[882e^{24x+34t}+196e^{10x+\frac{284}{3}t}+9e^{34x+\frac{2}{3}t}+109e^{14x+\frac{202}{3}t}\nonumber\\
&+452e^{20x+\frac{184}{3}t}+100e^{6x+122t}+450e^{28x+\frac{20}{3}t}+2e^{128t}],\\
Y&=e^{\frac{200}{3}t}+e^{20x}+98e^{10x+\frac{100}{3}t}+50e^{14x+6t}+50e^{6x+\frac{182}{3}t}.\end{aligned}$$ These solutions are finite and bounded. The graphs of the functions $|v_1|^2$ and $|v_2|^2$ are given in Figure 6(a) and Figure 6(b) respectively.

**ii.** **$v_4=a_1\bar{v}_2^{\varepsilon}$, $v_3=b_1\bar{v}_1^{\varepsilon}$**, $ \bar{v}_i^{\varepsilon}=\bar{v}_i(\varepsilon_1 t,\varepsilon_2 x)$, $\varepsilon_i^2=1$, $i= 1, 2$, $a_1$ and $b_1$ are constants. 