To compute the unique formal normal form of families of vector fields with nilpotent linear part, we choose a basis of the Lie algebra consisting of orbits under the action of the nilpotent linear part. This creates a new problem: to find explicit formulas for the structure constants in this new basis. These are well known in the 2D case, and recently expressions were found for the 3D case by ad hoc methods. The goal of the this paper is to formulate a systematic approach to this calculation. We propose to do this using a rational method for the inversion of the Clebsch–Gordan coefficients. We illustrate the method on a family of 3D vector fields and compute the unique formal normal form for the Euler family both in the 2D and 3D cases, followed by an application to the computation of the unique normal form of the Rössler equation.

中文翻译：

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多项式向量场的等变分解

为了计算具有幂零线性部分的向量场族的唯一形式范式，我们选择了一个李代数的基，该基由在幂零线性部分作用下的轨道组成。这就产生了一个新问题：在这个新的基础上找到结构常数的明确公式。这些在 2D 情况下是众所周知的，最近发现了 3D 情况下的表达式特设方法。本文的目标是制定一种系统化的计算方法。我们建议使用一种合理的方法来反演 Clebsch-Gordan 系数。我们在 3D 矢量场族上说明该方法，并在 2D 和 3D 情况下计算 Euler 族的唯一形式范式，然后将其应用于计算 Rössler 方程的唯一范式。