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

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.