On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of $\mathrm{C}$-Class

The fundamental invariants for vector ODEs of order $\ge 3$ considered up to point transformations consist of generalized Wilczynski invariants and $\mathrm{C}$-class invariants. An ODE of $\mathrm{C}$-class is characterized by the vanishing of the former. For any fixed $\mathrm{C}$-class invariant $\mathcal{U}$, we give a local (point) classification for all submaximally symmetric ODEs of $\mathrm{C}$-class with $\mathcal{U} \not \equiv 0$ and all remaining $\mathrm{C}$-class invariants vanishing identically. Our results yield generalizations of a well-known classical result for scalar ODEs due to Sophus Lie. Fundamental invariants correspond to the harmonic curvature of the associated Cartan geometry. A key new ingredient underlying our classification results is an advance concerning the harmonic theory associated with the structure of vector ODEs of $\mathrm{C}$-class. Namely, for each irreducible $\mathrm{C}$-class module, we provide an explicit identification of a lowest weight vector as a harmonic $2$-cochain.