Quadratic Poisson brackets for the Camassa--Holm peakons

We establish quadratic Poisson brackets for the generalized Camassa–Holm peakon structure introduced in \cite{AFR23}. The calculation is based on the halving of the spectral parameter dependent $r$-matrix used to define the linear Poisson structure of this model. This quadratic structure, together with the linear one, establish the bi-Hamiltonian structure of the generalized Camassa–Holm peakon model. \ When the deformation parameter tends to $\pm 2$, the spectral parameter dependence drops out, and we recover the linear and quadratic Poisson structure of the Camassa–Holm peakon model. \ When the spectral parameter tends to the fixed points of the involution defining the halving, we recover the Ragnisco–Bruschi deformation of the Camassa–Holm peakon model, thereby establishing its quadratic Poisson structure.