  The Camassa-Holm (CH) and Hunter-Saxton (HS) equations have an interpretation
as geodesic flow equations on the group of diffeomorphisms, preserving the
$H^1$ and $\dot{H}^1$ right-invariant metrics correspondingly. There is an
analogy to the Euler equations in hydrodynamics, which describe geodesic flow
for a right-invariant metric on the infinite-dimensional group of
diffeomorphisms preserving the volume element of the domain of fluid flow and
to the Euler equations of rigid body whith a fixed point, describing geodesics
for a left-invariant metric on SO(3). The CH and HS equations are integrable
bi-hamiltonian equations and one of their Hamiltonian structures is associated
to the Virasoro algebra. The parallel with the integrable SO(3) top is made
explicit by a discretization of both equation based on Fourier modes expansion.
The obtained equations represent integrable tops with infinitely many momentum
components.
  An emphasis is given on the structure of the phase space of these equations,
the momentum map and the space of canonical variables.
