the pair of the solutions of (\[TsymmcompN=4sys\]) becomes $$\displaystyle
|v_1|^2=\frac{81e^{2x}}{(4e^{3x}+9\cos(\frac{9}{2}t))^2+81\sin^2(\frac{9}{2}t)},\,
|v_2|^2=\frac{81e^{4x}}{(4e^{3x}+9\cos(\frac{9}{2}t))^2+81\sin^2(\frac{9}{2}t)}.$$ Both of the functions have singularity at $(t,x)=(\frac{(4n+2)}{9}\pi, x=\frac{2}{3}\ln (\frac{3}{2}))$. The graph of the solution is given in Figure 7.

![Singular solution for (\[TsymmcompN=4sys\]) with $\mu=2i, a_1=b_1=1, k_1=1, k_2=2, c_{12}=c_{13}=c_{23}=c_{24}=e^{\delta_1}=e^{\delta_2}=1$.](figurel.jpg)

b\) **S-Symmetric Nonlocal Complex HI-mKdV System:** $$\begin{aligned}
\label{SsymmcompN=4sys}
&\mu v_{1,t}(t,x)+6[c_{12}v_1(t,x)v_{2}(t,x)+c_{13}b_1v_1(t,x)\bar{v}_1(t,-x)+\bar{c}_{23}\bar{b}_1v_1(t,x)\bar{v}_2(t,-x)\nonumber\\
&+c_{23}b_1v_2(t,x)\bar{v}_1(t,-x)+c_{24}a_1v_2(t,x)\bar{v}_2(t,-x)+\bar{c}_{12}\bar{v}_1(t,-x)\bar{v}_2(t,-x)]v_{1,x}(t,x)\nonumber\\
&+v_{1,xxx}(t,x)=0,\nonumber\\
 &\mu v_{2,t}(t,x)+6[c_{12}v_1(t,x)v_{2}(t,x)+c_{13}b_1v_1(t,x)\bar{v}_1(t,-x)+\bar{c}_{23}\bar{b}_1v_1(t,x)\bar{v}_2(t,-x)\nonumber\\
&+c_{23}b_1v_2(t,x)\bar{v}_1(t,-x)+c_{24}a_1v_2(t,x)\bar{v}_2(t,-x)+\bar{c}_{12}\bar{v}_1(t,-x)\bar{v}_2(t,-x)]v_{2,x}(t,x)\nonumber\\
&+v_{2,xxx}(t,x)=0.\end{aligned}$$

**Example 8.** Take the following set of the parameters: $\mu=4i, a_1=b_1=1, k_1=\frac{i}{4}, k_2=\frac{i}{2}, c_{12}=c_{13}=c_{23}=c_{24}=e^{\delta_1}=e^{\delta_2}=1$. Hence the solutions of (\[SsymmcompN=4sys\]) become $$\displaystyle
|v_1|^2=\frac{81e^{\frac{1}{128}t}}{(64e^{\frac{9}{256}t}-9\cos(\frac{3}{4}x))^2+\sin^2(\frac{3}{4}x)}, \,|v_2|^2=\frac{81e^{\frac{1}{16}t}}{(64e^{\frac{9}{256}t}-9\cos(\frac{3}{4}x))^2+\sin^2(\frac{3}{4}x)}.$$ The solutions are finite for $t\geq 0$ and bounded for any $(x,t)$. The graph of the function $|v_1|^2$ is given in Figure 8.

![Periodic wave solution for (\[SsymmcompN=4sys\]) with $\mu=4i, a_1=b_1=1, k_1=\frac{i}{4}, k_2=\frac{i}{2}, c_{12}=c_{13}=c_{23}=c_{24}=e^{\delta_1}=e^{\delta_2}=1$.](figurem.jpg)

c\) **ST-Symmetric Nonlocal Complex HI-mKdV System:** $$\begin{aligned}
\label{STsymmcompN=4sys}
&\mu v_{1,t}(t,x)+6[c_{12}v_1(t,x)v_{2}(t,x)+c_{13}b_1v_1(t,x)\bar{v}_1(-t,-x)+\bar{c}_{23}\bar{b}_1v_1(t,x)\bar{v}_2(-t,-x)\nonumber\\
&+c_{23}b_1v_2(t,x)\bar{v}_1(-t,-x)+c_{24}a_1v_2(t,x)\bar{v}_2(-t,-x)+\bar{c}_{12}\bar{v}_1(-t,-x)\bar{v}_2(-t,-x)]v_{1,x}(t,x)\nonumber\\
&+v_{1,xxx}(t,x)=0,\nonumber\\
 &\mu v_{2,t}(t,x)+6[c_{12}v_1(t,x)v_{2}(t,x)+c_{13}b_1v_1(t,x)\bar{v}_1(-t,-x)+\bar{c}_{23}\bar{b}_1v_1(t,x)\bar{v}_2(-t,-x)\nonumber\\
&+c_{23}b_1v_2(t,x)\bar{v}_1(-t,-x)+c_{24}a_1v_2(t,x)\bar{v}_2(-t,-x)+\bar{c}_{12}\bar{v}_1(-t,-x)\bar{v}_2(-t,-x)]v_{2,x}(t,x)\nonumber\\
&+v_{2,xxx}(t,x)=0.\end{aligned}$$ **Example 9.** Let us take the parameters as $\mu=20, a_1=1, b_1=2i, k_1=-\frac{i}{2}, k_2=2i, c_{12}=c_{13}=c_{23}=i, c_{24}=1, e^{\delta_1}=20, e^{\delta_2}=2i$. In this case the solutions of (\[STsymmcompN=4sys\]) become $$\displaystyle
|v_1|^2=\frac{32400}{102481+5760\cos(\frac{3}{2}x+\frac{63}{160}t)}, \,|v_2|^2=\frac{324}{102481+5760\cos(\frac{3}{2}x+\frac{63}{160}t)}.$$ The above functions are finite and bounded for any $(x,t)$. The graph of the function $|v_1|^2$ is given in Figure 9.

![Periodic wave solution for (\[STsymmcompN=4sys\]) with $\mu=20, a_1=1, b_1=2i, k_1=-\frac{i}{2}, k_2=2i, c_{12}=c_{13}=c_{23}=i, c_{24}=1, e^{\delta_1}=20, e^{\delta_2}=2i$.](figuren.jpg)

Concluding Remarks
==================

In this work 