On the existence of Bertrand pairs (Q664197)

In the note of \textit{G. Saban} [``Barbier's theorem through self-mated closed Bertrand curves'', Simon Stevin 57, 37--45 (1983; Zbl 0512.53003)] the statememt is contained that for a simple closed twisted curve the Bertrand group would be trivial. Here the authors can show that for such a curve the Bertrand group can be either trivial or \(\mathbb Z_2 \). But they do not know whether this latter group can occur. Furthermore, for some classes of such curves it turns out that the non-triviality of those groups forces that the curve must be planar. In 6 propositions the authors prove several properties of such curves.