  The fundamental invariants for vector ODEs of order $\ge 3$ considered up to
point transformations consist of generalized Wilczynski invariants and C-class
invariants. 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 ${\mathcal
U} \not \equiv 0$ and all remaining 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 C-class. Namely, for each
irreducible C-class module, we provide an explicit identification of a lowest
weight vector as a harmonic 2-cochain.
