Communications in Analysis and Geometry

Volume 30 (2022)

Number 10

On $\mathrm{C}$-class equations

Pages: 2231 – 2266



Andreas Čap (Fakultät für Mathematik, Universität Wien, Austria)

Boris Doubrov (Faculty of Mathematics and Mechanics, Belarusian State University, Minsk, Belarus)

Dennis The (Department of Mathematics and Statistics, UiT, The Arctic University of Norway, Tromsø, Norway; and Fakultät für Mathematik, Universität Wien Austria)


The concept of a $\mathrm{C}$-class of differential equations goes back to E. Cartan with the upshot that generic equations in a $\mathrm{C}$-class can be solved without integration. While Cartan’s definition was in terms of differential invariants being first integrals, all results exhibiting $\mathrm{C}$-classes that we are aware of are based on the fact that a canonical Cartan geometry associated to the equations in the class descends to the space of solutions. For sufficiently low orders, these geometries belong to the class of parabolic geometries and the results follow from the general characterization of geometries descending to a twistor space.

In this article, we answer the question of whether a canonical Cartan geometry descends to the space of solutions in the remaining cases of scalar ODE of order at least four and of systems of ODE of order at least three. As in the lower order cases, this is characterized by the vanishing of the generalized Wilczynski invariants, which are defined via the linearization at a solution. The canonical Cartan geometries (which are not parabolic geometries) are a slight variation of those available in the literature based on a recent general construction. All the verifications needed to apply this construction for the classes of ODE we study are carried out in the article, which thus also provides a complete alternative proof for the existence of canonical Cartan connections associated to higher order (systems of) ODE.

The first and third authors were respectively supported by projects P27072-N25 and M1884-N35 of the Austrian Science Fund (FWF). D.T. was also supported by the Tromsø Research Foundation (project “Pure Mathematics in Norway”).

Received 25 January 2019

Accepted 22 July 2020

Published 29 September 2023