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

On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class