We associate an elliptic curve to each point of a natural parametrization of a spacelike naturally parametrized curve \(C\) in the Lorentz plane. The main tool of our study is the curvature of \(C\) and the equilateral hyperbola appears as a remarkable example since its elliptic curve is a CM one. Other two elliptic curves are obtained with this approach, the second one corresponding to the adjoint differential equation of the first, which in turn, is associated to a Lorentzian version of the logarithmic spiral.
Mircea Crasmareanu (Thu,) studied this question.