Periodic peakons and Calogero-Francoise flows

It has long been known that a number of periodic completely integrable systems are associated to hyperelliptic curves, for which the Abel map linearizes the flow (at least in part). We show that this is true for a relatively recent such system: the periodic discrete reduction of the shallow water equation derived by Camassa and Holm. The associated spectral problem has the same form and evolves in the same way as the spectral problem for a family of finite-dimensional non-periodic Hamiltonian flows introduced by Calogero and Francoise. We adapt the Weyl function method used earlier by us to solve the peakon problem to give an explicit solution to both the periodic discrete Camassa-Holm system and the (non-periodic) Calogero-Francoise system in terms of theta functions.