define the three conformal curvatures (Proposition \[propuniq\]). Then we use the variational equation to show that the curvatures are either constants or else can be expressed in terms of Jabobi’s elliptic functions (Proposition \[ConfCurv\]). In the first case the trajectory is an orbit of a $1$-parameter group of conformal transformations. Leaving aside the constant curvature case, we prove that the conformal equivalence classes of world lines can be parameterized by three real parameters (Proposition \[PHRS\]). Then we define the momentum operator, that is an element of the Lie algebra of the conformal group, intrinsically defined by the world-line. The existence of the momentum is a consequence of the conformal invariance of the functional and of the Nöther conservation theorem (Remark \[ThEx\]). We prove that the momentum operator is either a regular or an exceptional element of the conformal Lie algebra (Proposition \[CT\]). Then we define the integrating factors of an eigenvalue of the momentum and we build the principal vectors of the eigenvalues (Proposition \[PV\]). The explicit computation of the integrating factors is rather technical and is considered in the Appendix. In Section $5$ we use the integrating factors and the principal vectors to integrate by quadratures the trajectory of 