D_t+D_x^3)\{g_i\cdot f\}=0,\quad i=1, 2, \\
&D_x^2\{f\cdot f\}= 2c_{12}g_1g_2.\end{aligned}$$

One-Soliton Solution of N=2 Coupled HI-mKdV System
--------------------------------------------------

Here we will consider the solution given by (\[g\_if\]) with the pfaffian elements (\[pf1\])-(\[pf3\]) under the condition (\[pf4\]). The solution which has one-soliton for every $\displaystyle v_i=\frac{g_i}{f}$, $i=1, 2$, i.e., $M_1=1$, $M_2=1$ so $L=M_1+M_2=2$ with $B_1=\{b_1\}$ and $B_2=\{b_2\}$ is expressed by $$\begin{aligned}
&g_i=\mathrm{pf}(d_0,a_1,a_2,b_1,b_2,\beta_i),\, i=1,2,\\
&f=\mathrm{pf}(a_1,a_2,b_1,b_2),\end{aligned}$$ which are explicitly given as $$\begin{aligned}
& g_i=-e^{\theta_i},\quad \theta_i=k_ix-\frac{k_i^3}{a}t+\delta_i,\,\, i=1, 2,\\
&f=-1-\frac{c_{12}}{(k_1+k_2)^2}e^{\theta_1+\theta_2}.\end{aligned}$$ Hence one-soliton solution of the system (\[N=2a\])-(\[N=2b\]) is $$\displaystyle v_i=\frac{e^{\theta_i}}{1+\frac{c_{12}}{(k_1+k_2)^2}e^{\theta_1+\theta_2}},\,\, i=1,2.$$ Here $k_i, \delta_i$, $i=1, 2$ are arbitrary constants. The above solution is exactly the solution we obtained in [@GurPek2] for $c_{12}=-1$.

Local and Nonlocal Reductions for N=2
=====================================

The coupled HI-mKdV system for $N=2$ given by (\[N=2a\])-(\[N=2b\]) has four consistent reductions; two of them are local and the others are nonlocal. To obtain one-soliton solution of the reduced equations we use Type 1 and Type 2 approaches given in [@GurPek2]. Since the solutions for $N=2$ case are given in [@GurPek2] we will not present them here.

Local Reductions for N=2
------------------------

The coupled HI-mKdV system for $N=2$ given by (\[N=2a\])-(\[N=2b\]) has two local reductions. To obtain one-soliton solution of the reduced equations we use Type 1 