On Calogero-Francoise-type Lax matrices and their dynamical r-matrices

New classical integrable systems of Camassa-Holm peakon type are proposed. They realize the maximal even piecewise-D_2 generalization of the Calogero-Francoise flows, yielding periodic and pseudoperiodic trigonometric/hyperbolic potentials. The associated r-matrices are computed. They are dynamical and depend on both sets {p_i} and {q_i} of canonical variables.

💡 Research Summary

The paper introduces a new family of classical integrable many‑body systems that extend the well‑known Calogero‑Francoise (CF) flows by incorporating the maximal even piece of a (D_{2}) symmetry. The authors start by recalling the standard CF model, whose Lax matrix is built from canonical coordinates ({q_i}) and momenta ({p_i}) with a pairwise interaction potential of the form (1/\sinh^{2}(q_i-q_j)) (or its trigonometric counterpart). They then propose a generalized Lax matrix
\