On vector fields with simply connected trajectories and one invariant line
All polynomial vector fields on \(\mathbb{C}^2\) whose flows are complete (i.e., exist for all complex times) have been classified by \textit{M. Brunella} [Topology 43, No. 2, 433--445 (2004; Zbl 1047.32015)]. Up to polynomial automorphisms of \(\mathbb{C}^2\) they fall into three classes, where the first class is of the following form: \[ \Xi = (a x + b) \frac{\partial}{\partial x} + (A(x) y + B(x)) \frac{\partial}{\partial y} \] with \(a, b \in \mathbb{C}\) and polynomials \(A, B : \mathbb{C} \to \mathbb{C}\). The flow map of such a \(\Xi\) is easily seen to be complete, but in general not polynomial. The trajectories do not need to be simply connected. It is more convenient to interpret \(\Xi : \mathbb{C}[x,y] \to \mathbb{C}[x,y]\) as a \(\mathbb{C}\)-linear derivation. One can then distinguish two cases: {\parindent=6mm \begin{itemize}\item[(1)] If \(a = 0\), one may assume \(b = 1\) after another coordinate change in \(x\). The polynomial \(x\) is mapped to \(1\) by \(\Xi\). (If \(\Xi\) were a LND, such \(x\) would be called a slice.) \item[(2)] If \(a \neq 0\), one may assume \(b = 0\). The polynomial \(x\) is an eigenfunction of \(\Xi\) to the eigenvalue \(a \neq 0\). \end{itemize}} The authors now prove a partial converse under the extra assumption that the trajectories are simply connected: \smallskip Let \(\Xi : \mathbb{C}[x,y] \to \mathbb{C}[x,y]\) be a \(\mathbb{C}\)-linear derivation with an eigenfunction \(P\) to a non-zero eigenvalue, where \(P\) is assumed to be a primitive polynomial. If the trajectories of \(\Xi\) are simply connected, then it is complete. Moreover, up to a polynomial change of coordinates, it is of the above form -- with extra conditions to ensure that the trajectories are simply connected: \(a \neq 0\), \(b=0\), \(A(0) = 0\) and \(B(0) \neq 0\). This corresponds to case (2) above. Case (1) has been treated before by the same authors [Adv. Math. 285, 1339--1357 (2015; Zbl 1337.32034)].
scientific article; zbMATH DE number 6827610