we focus on the world-lines of $\mathcal{E}^{1,3}$ which are not trapped in a lower dimensional totally umbilical Einstein universe (linearly-full world lines). We prove that their trajectories can be explicitly integrated by quadratures in terms of elliptic functions and elliptic integrals.

The paper is organized as follows. In Section \[s1\], we collect from the literature few basic facts about conformal Lorentzian geometry [@BCDG; @GS] and we reformulate in the Lorentzian context the classical approach to the conformal geometry of curves in the Möbius space [@F; @Ha; @Mon; @SS; @S; @T; @Thomsen]. We define the conformal strain and the conformal line-element, which is the Lorentzian analogue of the conformal arc element of a curve in $S^n$ [@LO2010; @MMR; @Mon; @M1; @M2; @MN]. In Section \[s2\], we use the moving frame method to compute the Euler-Lagrange equations of the conformal strain functional (Theorem \[thmA\]). Consequently we show that the trajectory of a conformal world-line lies in a totally umbilical Einstein universe of dimension $2$, $3$ or $4$ (Theorem \[ThmB\]). In Section \[s3\] we study linearly full conformal world-lines of a four-dimensional Einstein universe. Given such a world-line we build a canonical lifting to the conformal group, the canonical conformal frame and we 