# Some remarkable autonomous systems

## TL;DR

This paper explores the mathematical structures connecting Darboux-Halphen-Ramanujan systems, contact geometry, Frobenius manifolds, and projective connections, emphasizing the role of contact geometry and autonomous systems derived from group determinants.

## Contribution

It establishes new links between complex geometric structures and autonomous systems, highlighting the significance of contact geometry in this context.

## Key findings

- Connections between Darboux-Halphen-Ramanujan systems and contact geometry
- Identification of canonical coordinates in Frobenius manifolds
- Analysis of autonomous systems from group determinants

## Abstract

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 $\Z/n\Z$, $n\geqslant3$, and the Klein group $\Z/2\Z\times\Z/2\Z$.