We show that the two orbits in the coadjoint representation of the group SO(4,2), the one, related with the Kepler's problem, and the other - modelling the phase space of a conformal particle with zero mass and spirality zero, coincide.
"The Phase Space for Conformal Particle with Zero Mass and Spirality and Its Relation with the Kepler's Problem." J. Geom. Symmetry Phys. 3 84 - 104, 2005. https://doi.org/10.7546/jgsp-3-2005-84-104