Inspired by [this question and the  counter example provided in its answer](https://mathoverflow.net/questions/293197/a-complex-limit-cycle-not-intersecting-the-real-plane) we ask:

>Is there a  polynomial vector  field  on $\mathbb{R}^2$ such that after complexification of the equation, the  corresponding singular holomorphic foliation of $\mathbb{C}^2$ possess a regular  complex leaf  $L$ whose holonomy is nontrivial  and $L$ does not intersect the real part  of $\mathbb{C}^2$? That is $L$ does not intersect $\{(z,w)\in \mathbb{C}^2 \mid im(z)=im (w)=0\}$.


**Note:** The  counter example in the above linked post shows that this  situation can occur if the  coefficient of polynomial vector  field are  complex. But what  about if the  coefficients are real?