ag.algebraic geometry - Spectral curve of Elliptic Calogero-Moser systems

So, the Lax operator $L(lambda)$ is given by
$$L(t,lambda)_{ij}=p_i delta_{ij}+(1-delta_{ij})f_{ij}Phi(q_i-q_j,lambda)$$
with lambda the spectral parameter, and $Phi$ the Lamé function. Using the Lax equation $dot{L}=[L,M]$, which is equivalent to $[L,frac{partial}{partial t}+M]=0$, if a matrix $A(t,lambda)$ satisfies $$left(frac{partial}{partial t}+M(t,lambda)right)A(t,lambda)=0$$ and is normalized, $A(0,lambda)=1$ it follows that
$$L(t,lambda)A(t,lambda)=A(t,lambda)L(0,lambda)$$
Hence, it is clear that $det(L-mu I)$, (and so the spectral curve) is independent of time. Now, the equation of the spectral curve is
$$Gamma:quaddet(L(t,lambda)-mu I)=0$$
Writing $$Gamma(lambda, mu)equivdet(L(t,lambda)-mu I)=sum_{i=0}^N r_i(lambda)mu^i$$
Your first question is why are the $r_i(lambda)$'s elliptic functions. Note that the matrix elements of $L$ are already doubly periodic, but they have an essential singularity at $lambda=0$. To show that the $r_i$'s are meromorphic, all you need is a gauge transformation to get rid of this singularity. Note that
$$L(t,lambda)=G(t,lambda)bar{L}(t,lambda)G^{-1}(t,lambda)$$
with $$G=left(delta_{ij}e^{zeta(lambda)q_i(t)}right)_{1le i,jle N}$$
where $zeta$ is the Weierstrass zeta function, does the job. So each $r_i(lambda)$ will be a combination of the Weierstrass $wp$ function and its derivatives, with the coefficients being integrals of the system. For each set of initial values of these integrals, the spectral curve is an $N$-sheeted covering of the base elliptic curve. The branch points will coincide with the zeros of $frac{partial Gamma(lambda,mu)}{partial lambda}$ on $Gamma$.

Look at "Introduction to classical integrable systems" by O. Babelon, D. Bernard, M. Talon, and the paper of Krichever I mention in the comments for more details.