Some remarkable autonomous systems

We study the links of the Darboux-Halphen-Ramanujan system, with contact geometry, canonical coordinates of some $3$-dimensional Frobenius manifolds and projective connections on Riemann surfaces. One of our important goals is to highlight the role of contact geometry in this setting. We also study autonomous systems “derived” from the potential given by the Group-determinant, of any cyclic group $\mathbb{Z}\mathrm{/}n\mathbb{Z}$, $n⩾3$, and the Klein group $\mathbb{Z}\mathrm{/}2\mathbb{Z}\times \mathbb{Z}\mathrm{/}2\mathbb{Z}$.