  We consider geodesics on the surfaces obtained by weak deformations of the
standard 2D-sphere. The dynamics of a particle on the surface can be
asymptotically described by the averaged evolution of the particle's angular
momentum. It is shown that the system describing this evolution has a
Hamiltonian, which is obtained by applying the Funk transform to the function
defining the deviation of the surface from the standard sphere. This system has
the 2D-sphere as its phase space, so it is integrable and its trajectories
admit of topological description in terms of its phase portrait on the sphere.
