yields the RS integrable system which is the relativistic analog of the trigonometric Calogero-Moser system.

On the open part $\bold{U}$ of $CM_\tau$ described in the subsection \[coor\] the Poisson bracket $\{\cdot,\cdot\}_{FR}$ takes the form (see Appendix of [@FR] for the proof): $$\begin{gathered}
\{\lambda_i,\lambda_j \}_{FR}=0,\quad
\{\lambda_j,q_i\}_{FR}=\lambda_i q_i\delta_{ij},\\
\{q_i,q_j\}_{FR}=
\frac{(\tau-\tau^{-1})^2q_iq_j(\lambda_i+\lambda_j)\lambda_i\lambda_j}
{(\tau\lambda_i-\tau^{-1}\lambda_j)(\tau\lambda_j-\tau^{-1}\lambda_i)
(\lambda_i-\lambda_j)}.\end{gathered}$$

The formulas in [@FR] contain the misprint, the authors lost the factor $(\tau^2-1)^2$ in the expression for $\{
q_i,q_j\}_{FR}$.

Using the Hamiltonian reduction on the combinatorial model of the space of flat connections on the torus without a point the authors of [@FR] prove that the Poisson structure $\{\cdot,\cdot\}_{FR}$ has a holomorphic extension from $\bold{U}$ to the whole $CM_\tau$, and this Poisson structure is nondegenerate (i.e. $CM_\tau$ is a symplectic variety). Another way to see this Poisson structure is to use Quasi-Poisson reduction [@AS]. In this picture the Poisson structure is the result of the reduction of the natural Quasi-Poisson structure on the product $GL(n,{\mathbb C})\times GL(n,{\mathbb C})$ and it is immediate that this Poisson structure is symplectic.

Finite dimensional representation of $H_{1,\tau}$
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In this subsection we construct a family of finite dimensional representations of $H_{1,\tau}$. Later we will show that this family forms an open dense set inside 