Force free projective motions of the sphere

Suppose that the sphere SnSn has initially a homogeneous distribution of mass and let GG be the Lie group of orientation preserving projective diffeomorphisms of SnSn. A projective motion of the sphere, that is, a smooth curve in GG, is called force free if it is a critical point of the kinetic energy functional. We find explicit examples of force free projective motions of SnSn and, more generally, examples of subgroups HH of GG such that a force free motion initially tangent to HH remains in HH for all time (in contrast with the previously studied case for conformal motions, this property does not hold for H=SOn+1H=SOn+1). The main tool is a Riemannian metric on GG, which turns out to be not complete (in particular not invariant, as happens with non-rigid motions), given by the kinetic energy.