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

TL;DR

The paper addresses local integrability and linearizability of 3D polynomial systems with a resonant diagonal linear part having eigenvalues $(1,\zeta,\zeta^2)$, establishing a convergence criterion for the $Poincaré\text{-}Dulac$ normal form and linking it to the existence of first integrals. It develops recursive formulas for integrability quantities $g_{kkk}$ and coefficients $v_{ijk}$, and introduces two efficient algorithms to compute these objects in $\mathbb{C}[a,b,c]$, with a Maple implementation and practical tests. A key result is that the condition $Y_1(w)+Y_2(w)+Y_3(w)\equiv 0$ ensures convergence of the distinguished normal form, tying normal-form theory directly to integrability. The authors then apply their framework to a quadratic subfamily, employing $\zeta$-reversible symmetry and elimination theory to identify nine irreducible integrability components, yielding linearizable cases (e.g., $J_1,J_4,J_5,J_8,J_9$) and conjecturally non-linearizable integrable cases (e.g., $J_2,J_3,J_6,J_7$). Overall, the work provides algorithmic tools and a structural classification that advance the understanding of integrability and linearizability in higher-dimensional polynomial systems.

Abstract

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, ζ, ζ^2 $, where $ζ$ is a primitive cubic root of unity. We 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, 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.

Theorems & Definitions (18)

• Theorem 1
• Theorem 2
• Lemma 1
• proof
• Theorem 3
• proof
• Theorem 4
• proof
• Definition 1
• Theorem 5