Edit: This is a real coefficient version of the current post.

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:

The error in Petrovski and Landis' proof of the 16th Hilbert problem

Edit: This is a real coefficient version of the current post.

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:

The error in Petrovski and Landis' proof of the 16th Hilbert problem

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:

The error in Petrovski and Landis' proof of the 16th Hilbert problem

Edit: This is a real coefficient version of the current post.

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:

The error in Petrovski and Landis' proof of the 16th Hilbert problem