On the 7th order ODE with submaximal symmetry

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 $Sp(4)/SL(2)$.

💡 Research Summary

The paper investigates the unique seventh‑order ordinary differential equation (ODE) that possesses a ten‑dimensional group of contact symmetries, i.e. the sub‑maximal symmetry case for order‑seven equations. Using Lie’s theory of contact transformations the authors first show that a generic seventh‑order ODE can have at most eleven contact symmetries, but the only equation attaining the next‑to‑maximal value (ten) is uniquely determined up to contact equivalence. This equation is invariant under a ten‑dimensional subgroup of the symplectic group (Sp(4)); the stabiliser of a generic solution is isomorphic to (SL(2)).

To obtain explicit solutions the authors reinterpret the ODE geometrically. They consider the space of 2‑jets ((x,y,p=y’,q=y’’)) as a projective three‑space (\mathbb{P}^3) equipped with the standard contact 2‑form (\omega = dp - q,dx). A solution curve of the ODE lifts to a rational curve (X(t)=