Let $X$ be a real analytic vector field defined on an open connected subset $U$ of $\mathbb{R}^n$. Let $p \in \mathbb{R}^n \setminus U$. Let $L$ be union of the flow lines $\ell$ of $X$ such that $p$ is in the closure (in $\mathbb{R}^n$) of $\ell$. Sometimes this is known as ''the set of separatrices convergent to $p$''.

Does the set $L$ have any kind of structure? Is it subanalytic, analytic? Or what other conditions should be added to $X$ or $U$ to know that this is the case?