Linearization Problem for a System of Two Second-Order ODEs via Cartan's Method: Branch I

Batoul M. Raddad; Ahmad Y. Al-Dweik; Marwan Aloqeili; F. M. Mahomed

Linearization Problem for a System of Two Second-Order ODEs via Cartan's Method: Branch I

General Mathematics 2026-07-18 v1

Authors: Batoul M. Raddad , Ahmad Y. Al-Dweik , Marwan Aloqeili , F. M. Mahomed

Abstract

Cartan's method classifies the class of linearizable system of two second-order ODEs into many branches. This paper investigates Branch I of the classification, characterized by a rank-one generalized Wilczynski invariant matrix and the vanishing of two relative invariants $K_1$ and $L_1$ . It is demonstrated that any linearizable system belonging to this branch admits an eight-dimensional Lie point symmetry algebra. The canonical form for this class is provided and the invariant characterizations based on the obtained rank-zero invariant coframe and the corresponding constant structure equations are established. Also, a systematic procedure for constructing the linearizing point transformation is derived. The theoretical results are illustrated by several examples.

Cite

@article{arxiv.2607.16840,
  title  = {Linearization Problem for a System of Two Second-Order ODEs via Cartan's Method: Branch I},
  author = {Batoul M. Raddad and Ahmad Y. Al-Dweik and Marwan Aloqeili and F. M. Mahomed},
  journal= {arXiv preprint arXiv:2607.16840},
  year   = {2026}
}