On the field analogue of elliptic spin Calogero-Moser model: Lax pair and equations of motion

The Lax pair for the field analogue of the classical spin elliptic Calogero-Moser is proposed. Namely, using the previously known Lax matrix we suggest an ansatz for the accompany matrix. The presented construction is valid when the matrix of spin variables $\mathcal{S}\in \mathrm{M}\mathrm{a}\mathrm{t}(N,\mathbb{C})$ satisfies the condition ${\mathcal{S}}^2=c_0\mathcal{S}$ with some constant $c_0\in \mathbb{C}$. It is proved that the Lax pair satisfies the Zakharov-Shabat equation with unwanted term, thus providing equations of motion on the unreduced phase space. The unwanted term vanishes after additional reduction. In the special case $\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{k}(\mathcal{S})=1$ we show that the reduction provides the Lax pair of the spinless field Calogero-Moser model obtained earlier by Akhmetshin, Krichever and Volvovski.