The purpose of this paper is to provide results like Peixoto’s Theorem inside a class $\mathfrak{X}$ of 3-dimensional reversible systems. We deal with reversible systems associated to an involution $\varphi_1$ such that $\operatorname{Dim}\operatorname{Fix}(\varphi_1)$ is equal to one. We further assume the existence of a first integral for any $X\in\mathfrak{X}$ that leaves invariant any sphere centered at the origin. The main result establishes generic necessary conditions for the structural stability in $\mathfrak{X}$. Stability conditions on generic one-parameter families of reversible vector fields on the two-dimensional sphere are also stated.

中文翻译：

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可逆和部分可积系统的稳定性条件

本文的目的是在 3 维可逆系统的类 $\mathfrak{X}$ 中提供类似 Peixoto??s Theorem 的结果。我们处理与对合 $\varphi_1$ 相关的可逆系统，使得 $\operatorname{Dim}\operatorname{Fix}(\varphi_1)$ 等于 1。我们进一步假设任何 $X\in\mathfrak{X}$ 都存在第一积分，它使任何以原点为中心的球体保持不变。主要结果为 $\mathfrak{X}$ 中的结构稳定性建立了通用的必要条件。还给出了二维球面上可逆矢量场的泛型一参数族的稳定性条件。