Geodesic
Manifold Ways to Darboux–Halphen System
Symmetry, integrability and geometry: methods and applications, Tome 14 (2018) Cet article a éte moissonné depuis la source Math-Net.Ru

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Many distinct problems give birth to Darboux–Halphen system of differential equations and here we review some of them. The first is the classical problem presented by Darboux and later solved by Halphen concerning finding infinite number of double orthogonal surfaces in $\mathbb{R}^3$. The second is a problem in general relativity about gravitational instanton in Bianchi IX metric space. The third problem stems from the new take on the moduli of enhanced elliptic curves called Gauss–Manin connection in disguise developed by one of the authors and finally in the last problem Darboux–Halphen system emerges from the associative algebra on the tangent space of a Frobenius manifold.
Keywords: Darboux–Halphen system; Ramanujan system; Gauss–Manin connection; relativity and gravitational theory; Bianchi IX metric; Frobenius manifold; Chazy equation. 
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John Alexander Cruz Morales; Hossein Movasati; Younes Nikdelan; Raij Roychowdhury; Marcus A. C. Torres. Manifold Ways to Darboux–Halphen System. Symmetry, integrability and geometry: methods and applications, Tome 14 (2018). http://geodesic.mathdoc.fr/item/SIGMA_2018_14_a2/

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