On local integrability conditions for nowhere-zero complex vector fields (Q1326729)

Let \(X\) be a nowhere-zero complex vector field, with \(C^ \infty\) coefficients, in an open set in \(\mathbb{R}^{n+1}\). We say that \(X\) is locally integrable at a point \(P\) if the homogeneous equation \(Xu = 0\) has \(C^ 1\) solutions \(u_ 1, \dots, u_ n\) in a neighborhood \(U\) of \(P\) such that \(du_ 1 \wedge du_ 2 \wedge \cdots \wedge du_ n \neq 0\) in \(U\). The paper gives some local integrability conditions at points \(P\) on the \(x\)-axis for the vector field \(L: = \partial/ \partial t + ia(t,x) \cdot \partial/ \partial x\), where \(a(t,x)\) is a real valued \(C^ \infty\) function having the property \(ta(t,x)>0\) for \(t \neq 0\).