  We consider a family of integro-differential equations depending upon a
parameter $b$ as well as a symmetric integral kernel $g(x)$. When $b=2$ and $g$
is the peakon kernel (i.e. $g(x)=\exp(-|x|)$ up to rescaling) the
dispersionless Camassa-Holm equation results, while the Degasperis-Procesi
equation is obtained from the peakon kernel with $b=3$. Although these two
cases are integrable, generically the corresponding integro-PDE is
non-integrable. However,for $b=2$ the family restricts to the pulson family of
Fringer & Holm, which is Hamiltonian and numerically displays elastic
scattering of pulses. On the other hand, for arbitrary $b$ it is still possible
to construct a nonlocal Hamiltonian structure provided that $g$ is the peakon
kernel or one of its degenerations: we present a proof of this fact using an
associated functional equation for the skew-symmetric antiderivative of $g$.
The nonlocal bracket reduces to a non-canonical Poisson bracket for the peakon
dynamical system, for any value of $b\neq 1$.
