Darboux integrability for polynomial vector fields invariant under action of finite group (Q1936484)

It is clear that a polynomial vector field of degree \(d\) in \(K^n\), with \(K \, = \, \mathbb{R}\) or \(K \, = \, \mathbb{C}\), admits a rational first integral if and only if it possesses an infinite number of invariant irreducible algebraic hypersurfaces. [\textit{J. P. Jouanolou}, Equations de Pfaff algébriques. Lecture Notes in Mathematics. 708. Berlin-Heidelberg-New York: Springer-Verlag (1979; Zbl 0477.58002)] showed that, if the number of invariant irreducible hypersurfaces is finite, then it is bounded by \[ \left( \begin{matrix} d+n-1 \\ n \end{matrix}\right) + n. \] The present paper deals with polynomial vector fields which are indeed invariant under the action of a finite group \(G\). Denote by \(K_r[z]^G\) the space of \(G\)-invariant polynomials of degree \(r\). The main results in the paper is the following statement. If a \(G\)-invariant polynomial vector field admits \[ \dim_{K} \left( K_{d-1}[z]^G\right) + n \] invariant irreducible algebraic \(G\)-hypersurfaces, then it admits a \(G\)-rational first integral.