Hamiltonian for the field spin Calogero–Moser model

Determine the Hamiltonian functional for the 1+1 dimensional field analogue of the spin elliptic Calogero–Moser model of gl_N type, with spin matrix S(x) satisfying S^2 = c0 S and Lax pair given by U_{2dSpin}(z) in equation (4.1) and V_{2dSpin}(z) in equation (5.13), such that the corresponding Zakharov–Shabat equation generates the equations of motion (5.25)–(5.26).

Background

The paper constructs a Lax U–V pair and derives equations of motion for the field analogue of the spin elliptic Calogero–Moser model on the unreduced phase space, under the condition S^2 = c0 S. For the spinless field model, the Hamiltonian is known and presented in Section 3, but for the spin field case the authors proceed without giving a Hamiltonian, relying instead on the equations of motion and reduction arguments to connect to the spinless case.

In Section 6, when performing the reduction to the spinless field Calogero–Moser model, the authors explicitly state that the Hamiltonian for the field spin case is not known, and consequently they use the equations of motion to carry out the reduction. This indicates an unresolved problem of identifying a Hamiltonian that reproduces the derived field spin equations via the Zakharov–Shabat structure.