A remark on first integrals of vector fields
Andr\'e Belotto da Silva, Martin Klimes, Julio C. Rebelo, Helena Reis
TL;DR
This paper presents examples of vector fields in complex three-dimensional space that have formal first integrals but lack holomorphic ones, addressing a question in the theory of dynamical systems.
Contribution
It provides explicit examples linking formal and holomorphic integrability, expanding understanding of integrals in complex vector fields.
Findings
Existence of vector fields with formal but no holomorphic first integrals
Connection to longstanding questions in complex dynamical systems
Insights related to Malgrange and Mattei-Moussu theorems
Abstract
We provide examples of vector fields on admitting a formal first integral but no holomorphic first integral. These examples are related to a question raised by D. Cerveau and motivated by the celebrated theorems of Malgrange and Mattei-Moussu.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Advanced Algebra and Geometry · Holomorphic and Operator Theory