u>0\\
r,m>0
\end{array}}(\psi_{-u}^{2}\psi_{m+u}^{2*}-\psi_{m-u}^{2*}\psi_{u}^{2})(\psi_{-r}^{1}\psi_{r-m}^{1*}+\psi_{-m-r}^{1*}\psi_{r}^{1})\right)\,.\end{aligned}$$ Similarly, we have $$\begin{aligned}
H_{int}^{2} & = & -\frac{2r}{\sqrt{p}}\sum_{\begin{array}[t]{c}
n,m>0\end{array}}a_{-m}^{1}a_{-n}^{2}a_{n+m}^{2}\\
 & = & -\frac{2r}{\sqrt{p}}\left(\sum_{\begin{array}[t]{c}
r,s,k,l,u\in\mathbb{Z}+\frac{1}{2}\\
r+s+k+l=m>0
\end{array}}:\psi_{r}^{2}\psi_{s}^{2*}\psi_{k}^{2}\psi_{l}^{2*}\psi_{-u}^{1}\psi_{-m+u}^{1*}:\right.\\
 &  & \hspace{1em}\hspace{1em}+\left.\sum_{u\in\mathbb{Z}+\frac{1}{2},m>0}:\psi_{-u}^{1}\psi_{-m+u}^{1*}:(\text{Contractions})\right)\\
 &  & =-\frac{2r}{\sqrt{p}}\left(\sum_{\begin{array}[t]{c}
r,s,k,l,u\in\mathbb{Z}+\frac{1}{2}\\
r+s+k+l=m>0
\end{array}}:\psi_{r}^{2}\psi_{s}^{2*}\psi_{k}^{2}\psi_{l}^{2*}\psi_{-u}^{1}\psi_{-m+u}^{1*}:\right.\\
 &  & \hspace{1em}\hspace{1em}+\left.\sum_{\begin{array}[t]{c}
u,m>0\\
r>0
\end{array}}(\psi_{-u}^{1}\psi_{-m+u}^{1*}-\psi_{-m-u}^{1*}\psi_{u}^{1})(\psi_{-r}^{2}\psi_{r+m}^{2*}+\psi_{m-r}^{2*}\psi_{r}^{2})\right)\,.\end{aligned}$$ Though this fermionic formalism is not effective in calculating the eigenstate, it plays an important role in deriving the behind tau-function and its Hirota integrability of this system. Actually, the fermionic formalism completely defines the fermionic orbit of a generator of $GL(\infty)$ which in turn determine the tau-function of this theory. We are working in detail in this direction.

Similarity transformation
-------------------------

As explained before, we need to do a similarity transformation to recover the deformed operator formalism of eigenstates to a standard operator formalism. It is easy to write down this similarity transformation, that is $$D^{Hal}=D^{Lau}(b^{2}\rightarrow p,\, a\rightarrow a^{1})D^{Lau}(b^{2}\rightarrow q,\, a\rightarrow a^{2})\,.$$

Conclusions and Future Works
============================

In conclusion, we introduce a systematic way to extract the integrability of several models. For CS model, we express the Hamiltonian in bosonic as well as fermionic representations. The eigenstate and eigenvalue are obtained explicitly. The construction of Jack state, in the fermionic representation, is highly involved in the fermionic triangularization of fermionic Hamiltonian of the CS 