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 ${\text{PP}}^3$ which give rise to rational plane curves of degree six. The moduli space of these curves is a real form of the homogeneous space $Sp(4)\mathrm{/}SL(2)$.