Let $X$ be an analytic vector field on a compact oriantable surface $S$ with volume form $\omega$. We denote the set of its singularities by $Z(X)$.

**A local question**

Is there an analytic vector field $Y$ on a neighborhood of $Z(X)$ such that $Z(Y)\subset Z(X)$ and $Z(Y)$ is a finite set. Moreover $\omega(Y(p),X(p))=0$ for all $p\in S$. That is: $X \parallel Y$ out of singularities of $X$?

**A global question**

Can we find an analytic vector field $Y$ as above, globally on whole $S$?

**Motivation:** I think the **second** question is implicitly used (and is needed) in the book "Finiteness theorem for Limit cycles" by YU.S. Ilyashenko.

In Fact my main motivation is the following question:

**Main motivating question:**

Is it easy to pass from statement $A$ to $B$ as follows?:

**A:** Every analytic vector field $X$ on $S^{2}$ has a finite number of limit cycles provided the singular set of $X$ is a finite set.

**B:** Every arbitrary analytic vector field on $S^{2}$ has a finit number of limit cycles.

**Edit and update:** There is a new version of "Finitness theorem of limit cycle whose abstract indicates to a new version of the proof of finitness theorem for analytic vector fields.

http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=im&paperid=8352&option_lang=eng