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

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2023
• Main Authors: Kessy, Johnson Allen, The, Dennis
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2023
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211971
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: On Uniqueness of Submaximally Symmetric Vector Ordinary Differential Equations of C-Class. Johnson Allen Kessy and Dennis The. SIGMA 19 (2023), 058, 29 pages

The fundamental invariants for vector ODEs of order ≥ 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 𝒰, we give a local (point) classification for all submaximally symmetric ODEs of C-class with 𝒰 ≢ 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.