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<p>The authors 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, ζ, ζ<sup>2</sup>, where ζ is a primitive cubic root of unity. The authors establish a criterion for the convergence of the Poincaré-Dulac normal form of the systems and examine the relationship between the normal form and integrability. Additionally, the authors 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. The findings offer insights into the integrability properties of three-dimensional polynomial systems.</p>

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Integrability and Linearizability of a Family of Three-Dimensional Polynomial Systems

  • 期刊论文

  • 发表日期：

    2026年3月24日

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  • Bo Huang,
  • Ivan Mastev,
  • Valery G. Romanovski

摘要

The authors 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, ζ, ζ2, where ζ is a primitive cubic root of unity. The authors establish a criterion for the convergence of the Poincaré-Dulac normal form of the systems and examine the relationship between the normal form and integrability. Additionally, the authors 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. The findings offer insights into the integrability properties of three-dimensional polynomial systems.