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 $\pm2$, 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.