Algebraic integrability of the classical XXZ spin chain with reflecting boundary conditions

In this paper we analyze the classical XXZ spin chain with reflecting boundaries. We exhibit a system of log-canonical coordinates on the phase space generalizing Sklyanin's separation of variables for the periodic XXZ chain, and use these coordinates to construct action-angle variables for the system. We also integrate the flows of the reflection Hamiltonians explicitly in terms of Riemann theta functions. Central to our analysis is the algebraic integrability of the model.