  We study geodetic lines on a surface generated by a small deformation of the
standard 2D-sphere. We construct an auxiliary hamiltonian system with the view
of describing geodetic coils and almost closed geodesics, by using the fact
that loops of the coil can be well approximated by great circles of the sphere.
The phase space of the auxiliary system is determined by the graph generated by
separatrixes of its solutions, the vertices of the graph corresponding to
almost closed geodesics and the edges to the geodetic coils joining them.
Topological types of the graph depend on the parameters determining the
deformation. Using the method of averaging in conjunction with the computer
modelling of the auxiliary system, we obtain a fairly detailed visualization of
geodesics on the deformed sphere.
