We investigate the local integrability and linearizability of a family of three-dimensional polynomial systems with the matrix of the linear approximation having the eigenvalues $1, \zeta, \zeta^2 $, where $\zeta$ is a primitive cubic root of unity. We establish a criterion for the convergence of the Poincar\'e--Dulac normal form of the systems and examine the relationship between the normal form and integrability. Additionally, we introduce an efficient algorithm to determine the necessary conditions for the integrability of the systems. This algorithm is then applied to a quadratic subfamily of the systems to analyze its integrability and linearizability. Our findings offer insights into the integrability properties of three-dimensional polynomial systems.

翻译：本文研究了一类三维多项式系统的局部可积性与可线性化性，其线性近似矩阵的特征值为 $1, \zeta, \zeta^2 $，其中 $\zeta$ 为单位的三次本原根。我们建立了该系统 Poincaré--Dulac 范式收敛性的判别准则，并探讨了范式与可积性之间的关系。此外，我们提出了一种高效算法来确定系统可积性的必要条件。该算法随后被应用于系统的二次子族，以分析其可积性与可线性化性。我们的研究结果为理解三维多项式系统的可积性性质提供了新的见解。