  We show that the following three systems related to various hydrodynamical
approximations: the Korteweg--de Vries equation, the Camassa--Holm equation,
and the Hunter--Saxton equation, have the same symmetry group and similar
bihamiltonian structures. It turns out that their configuration space is the
Virasoro group and all three dynamical systems can be regarded as equations of
the geodesic flow associated to different right-invariant metrics on this group
or on appropriate homogeneous spaces. In particular, we describe how Arnold's
approach to the Euler equations as geodesic flows of one-sided invariant
metrics extends from Lie groups to homogeneous spaces. We also show that the
above three cases describe all generic bihamiltonian systems which are related
to the Virasoro group and can be integrated by the translation argument
principle: they correspond precisely to the three different types of generic
Virasoro orbits.
