**Edit: [This is a real coefficient version of the current post.](https://mathoverflow.net/questions/327721/a-complex-limit-cycle-not-intersecting-the-real-plane2?noredirect=1&lq=1)** Is there a polynomial vector field $X$ with complex coefficients on $\mathbb{C}^2$ with the property quoted bellow? >There is a regular leaf $L$ whose holonomy, along at least one closed curve on it, is not trivial but $L$ does not intersect the real part $im (z)=im(w)=0,\;(z,w) \in \mathbb{C}^2$. # Note: A leaf with non trivial holonomy is called a complex limit cycle, according to the terminology used in the video lecture by Ilyashenko described in the following answer: https://mathoverflow.net/questions/171988/the-error-in-petrovski-and-landis-proof-of-the-16th-hilbert-problem/173609#173609