On a property of Nirenberg type operator (Q1293340)

The author gives examples of \(C^\infty\) complex vector fields \(L\) in \(\mathbb{R}^2\), such that the equation \(Lu=0\) does not admit solutions \(u\in C^1\) in a neighborhood of the origin satisfying \(du(0)\neq 0\). The results in the present paper give a more general frame to a similar counterexample of \textit{L. Nirenberg} [Lectures on linear partial differential equations, Reg. Con. Ser. Math. No. 17, Am. Math. Soc. (1973; Zbl 0267.35001)].