On the 7th order ODE with submaximal symmetry

TL;DR

This paper derives a general solution for a specific 7th order ODE with maximal contact symmetry, linking it to rational contact and plane curves, and explores its moduli space structure.

Contribution

It provides the explicit general solution for a 7th order ODE with 10-dimensional contact symmetry group and characterizes its geometric and moduli space properties.

Findings

Solution describes rational contact curves in projective space

Connection to rational plane curves of degree six

Moduli space identified as a real form of Sp(4)/SL(2)

Abstract

We find a general solution to the unique 7th order ODE admitting ten dimensional group of contact symmetries. The integral curves of this ODE are rational contact curves in $\PP^3$ which give rise to rational plane curves of degree six. The moduli space of these curves is a real form of the homogeneous space $Sp(4)/SL(2)$.