On the field analogue of the elliptic spin Calogero–Moser model: lax pair and equations of motion

A Lax pair for the field analogue of the classical spin elliptic Calogero–Moser model is proposed. Namely, using the previously known Lax matrix, we suggest an ansatz for the accompanying matrix. The presented construction is valid when the matrix of spin variables ${\mathcal S}\in\operatorname{Mat}(N,\mathbb C)$ satisfies the condition ${\mathcal S}^2=c_0{\mathcal S}$ with some constant $c_0\in\mathbb C$. It is shown 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 $\operatorname{rank}(\mathcal S)=1$, we show that the reduction provides the Lax pair of the spinless field Calogero–Moser model obtained earlier by Akhmetshin, Krichever, and Volvovski.