a linearly full conformal world-line with non-constant curvatures (Theorem \[ThmC\] and Theorem \[ThmD\]). At the end of the section we briefly comment the theoretical aspects underlying the integration by quadratures, we explain why a linearly full world-line with non-constant curvatures, can’t be closed and we show that the trajectory of a world-line is invariant by the action of an infinite cyclic group of conformal transformations.

Conformal geometry of a time-like curve {#s1}
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The Einstein universe and its restricted conformal group
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We denote by $\mathcal{E}^{1,n-1}$ the n-dimensional sub-manifold of $\R^{n+2}, n>1$, defined by the equations $x_0^2+x_1^2=1$ and $x_2^2+\dots + x_{n+1}^2=1$. As a manifold, $\mathcal{E}^{1,n-1}$ is the Cartesian product $S^1 \times S^{n-1}$. The restriction of the quadratic form $$g=-dx_0^2-dx_1^2+dx_2^2+\dots + dx_{n+1}^2$$ induces a Lorentzian pseudo-metric $g_{\textsl{e}}$ on $\mathcal{E}^{1,n-1}$. The normal bundle of $\mathcal{E}^{1,n-1}$ is spanned by the restrictions of the vector fields $\mathbf{n}_1=x_0\partial_{x_0}+x_1\partial_{x_1}$ and $\mathbf{n}_2=x_2\partial_{x_2}+\dots +x_{n+1}\partial_{x_{n+1}}$. Thus, contracting $dx_0\wedge\dots dx_{n+1}$ with $\mathbf{n}_1$ and $\mathbf{n}_2$, we get a volume form on $\mathcal{E}^{1,n-1}$ which in turn defines an orientation. The vector field $-x_1\partial_{x_0}+x_0\partial_{x_1}$ is tangent to $\mathcal{E}^{1,n-1}$ and induces a unit time-like vector field on $\mathcal{E}^{1,n-1}$. We time-orient $\mathcal{E}^{1,n-1}$ by requiring that such a vector field is future-oriented.

The Lorentzian manifold $(\mathcal{E}^{1,n-1},g_{\textsl{e}})$, 