AI 中文总结
研究二阶常微分方程组线性化问题的分支I，利用嘉当方法，证明该分支可线性化系统有八维李点对称代数，给出标准形式、不变特征描述及线性化点变换构造程序，并通过实例说明理论结果。
AI 中文摘要
嘉当方法将二阶常微分方程组的可线性化系统分类为多个分支。本文研究分类中的分支I，其特征为一阶广义威尔钦斯基不变矩阵以及两个相对不变量$K_1$和$L_1$为零。证明了该分支的任何可线性化系统都有一个八维李点对称代数。给出了此类的标准形式，并基于得到的零秩不变余标架和相应的常数结构方程建立了不变特征描述。还推导了构造线性化点变换的系统程序。通过几个例子说明了理论结果。
英文摘要
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.
