On the relations between several notions of symmetry for the second order linear differential equation

Abstract

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 $W\mapstoμ^{-1}(W\circ\varphi)$ for characteristics under a contact transformation with conformal factor $μ$, which governs the transport of evolutionary representatives.

Disclosure

“3 teoría de Picard-Vessiot con la geometría y la dinámica” of Universidad Nacional de Colombia, code Hermes 67312. Use of AI-Assisted Tools. During the preparation of this work, the authors employed Claude Opus (Anthropic) and Gemini 3.7 (Google) to audit and verify the results from [1]. Insights from these tools led to the correction of minor inaccuracies, the restructuring of select proofs, and the refinement of theoretical statements as shown in the actual version of th”

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2026-08-23T05:21:38.755312Z

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