  We consider a new partial differential equation, of a similar form to the
Camassa-Holm shallow water wave equation, which was recently obtained by
Degasperis and Procesi using the method of asymptotic integrability. We prove
the exact integrability of the new equation by constructing its Lax pair, and
we explain its connection with a negative flow in the Kaup-Kupershmidt
hierarchy via a reciprocal transformation. The infinite sequence of conserved
quantities is derived together with a proposed bi-Hamiltonian structure. The
equation admits exact solutions in the form of a superposition of
multi-peakons, and we describe the integrable finite-dimensional peakon
dynamics and compare it with the analogous results for Camassa-Holm peakons.
