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Inspired by this question and the counter example provided in its answer 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?

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  • $\begingroup$ This case is the same as your example in the complex version of the question. Just consider $$z′=w+(z^2+w^2+4) \\ w′=−z+(z^2+w^2+4)$$. $\endgroup$ – Loïc Teyssier 12 hours ago

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