Invariant hyperplanes and Darboux integrability for \(d\)-dimensional polynomial differential systems (Q1841207)

This paper treats the following question: how many invariant hyperplanes can have a polynomial system of degree \(m=(m_1,\dots,m_d)\) for the subclass of regular polynomial systems in \(\mathbb{R}^d\)? Using Darboux theory of integrability the authors analyze when it is possible to find a first integral of a polynomial vector field of degree \((m_1,\dots,m_d)\) in \(\mathbb{R}^d\) by knowing the existence of a sufficient number of invariant hyperplanes.