  We propose realizations of the Poisson structures for the Lax representations
of three integrable $n$-body peakon equations, Camassa--Holm,
Degasperis--Procesi and Novikov. The Poisson structures derived from the
integrability structures of the continuous equations yield quadratic forms for
the $r$-matrix representation, with the Toda molecule classical $r$-matrix
playing a prominent role.
  We look for a linear form for the $r$-matrix representation. Aside from the
Camassa--Holm case, where the structure is already known, the two other cases
do not allow such a presentation, with the noticeable exception of the Novikov
model at $n=2$.
  Generalized Hamiltonians obtained from the canonical Sklyanin trace formula
for quadratic structures are derived in the three cases.
