AI 中文总结
本文研究二阶线性微分方程的多种无穷小对称概念，构建前三者的显式对应，推导切触对称李代数的分解与李括号公式，给出切触对称的等价刻画及特征变换律，明确分次分解无法覆盖完整对称代数。
AI 中文摘要
在二阶线性微分方程的研究文献中，存在多种不等价的无穷小对称性概念：Lie点对称、垂直（规范）对称、算子对称、无穷小切触对称，以及Lie–Bäcklund算子。我们构建了前三者之间的显式对应关系。随后，我们描述了切触系统$\Omega=\langle dy-y'\\,dx\rangle$的无穷小切触变换李代数$\mathcal L_\Omega(\mathbb U)$，其系数取自以$x$和$y$为变量的函数构成的微分域$\mathbb U$。我们得到其典范分解$\mathcal L_\Omega(\mathbb U)=\prod_{k\ge0}\mathcal L_\Omega^k\mathbb U$，分解为若干$\mathbb C$向量空间，每个空间由$\mathbb U$参数化（$k=0$时由$\mathbb U\oplus\mathbb U$参数化），并计算了这些坐标下李括号的代数微分公式。\n应用于对称性问题时，我们证明：生成函数为$W$的切触向量场是方程的对称，当且仅当$A^{2}W=aW+b\\,AW$，其中$A$是射流空间中对应于该方程的向量场；等价地，当且仅当$W=F_1(u_1,u_2)\phi_1+F_2(u_1,u_2)\phi_2$，其中$F_1,F_2$是$A$的两个首次积分的任意函数，$\phi_1,\phi_2$是解空间的一组基。该对称代数始终由两个二元任意函数参数化。研究还表明，$\mathcal L_\Omega(\mathbb U)$的分解永远无法覆盖整个对称代数：当$a\neq0$时，分次部分退化为点对称；而对于方程$y''=0$，分次部分是无限维但仍为真子空间，且两种情况下它都不是李子代数。最后，我们明确了具有共形因子$\mu$的切触变换下特征量的变换律$W\mapsto\mu^{-1}(W\circ\varphi)$，该变换律支配演化代表元的输运。
英文摘要
There are several non-equivalent notions of infinitesimal symmetry in the literature of second order linear differential equations: Lie point symmetries, vertical (gauge) symmetries, operator symmetries, infinitesimal contact symmetries, and Lie--Bäcklund operators. We construct an explicit correspondence among the first three, We then describe the Lie algebra $\mathcal L_Ω(\mathbb U)$ of infinitesimal contact transformations of the contact system $Ω=\langle dy-y'\,dx\rangle$, with coefficients in a differential field $\U$ of functions of $x$ and $y$. We obtain a canonical decomposition $\mathcal L_Ω(\mathbb U)=\prod_{k\ge0}\mathcal L_Ω^k\mathbb U$ into $\mathbb C$-vector spaces, each parametrized by $\mathbb U$ (by $\mathbb U\oplus\mathbb U$ for $k=0$), and we compute the algebraic differential formulae for the Lie bracket in these coordinates. Applied to the symmetry problem, we prove that a contact vector field with generating function $W$ is a symmetry of if and only if $A^{2}W=aW+b\,AW$, where $A$ is the vector field in the jet space corresponding to the equation; equivalently, if and only if $W=F_1(u_1,u_2)ϕ_1+F_2(u_1,u_2)ϕ_2$ for arbitrary functions $F_1,F_2$ of the two first integrals of $A$ and a basis $ϕ_1,ϕ_2$ of solutions. The symmetry algebra is always parametrized by two arbitrary functions of two variables. It also shows that the decomposition of $\Lom(\U)$ never captures the whole symmetry algebra: for $a\neq0$ the graded part reduces to the point symmetries, while for $y''=0$ it is an infinite dimensional but still \emph{proper} subspace, and in neither case is it a Lie subalgebra. Finally we make precise the transformation law $W\mapstoμ^{-1}(W\circφ)$ for characteristics under a contact transformation with conformal factor $μ$, which governs the transport of evolutionary representatives.
