  This paper is devoted to periodic travelling waves solving Lie-Poisson
equations based on the Virasoro group. We show that the reconstruction of any
such solution can be carried out exactly, regardless of the underlying
Hamiltonian (which need not be quadratic), provided the wave belongs to the
coadjoint orbit of a uniform profile. Equivalently, the corresponding "fluid
particle motion" is integrable. Applying this result to the Camassa-Holm
equation, we express the drift of particles in terms of parameters labelling
periodic peakons and exhibit orbital bifurcations: points in parameter space
where the drift velocity varies discontinuously, reflecting a sudden change in
the topology of Virasoro orbits.
