  We consider a two-component Hamiltonian system of partial differential
equations with quadratic nonlinearities introduced by Popowicz, which has the
form of a coupling between the Camassa-Holm and Degasperis-Procesi equations.
Despite having reductions to these two integrable partial differential
equations, the Popowicz system itself is not integrable. Nevertheless, as one
of the authors showed with Irle, it admits distributional solutions of peaked
soliton (peakon) type, with the dynamics of $N$ peakons being determined by a
Hamiltonian system on a phase space of dimension $3N$. As well as the trivial
case of a single peakon ($N=1$), the case $N=2$ is Liouville integrable. We
present the explicit solution for the two-peakon dynamics, and describe some of
the novel features of the interaction of peakons in the Popowicz system.
