**Edit:** According to the comment of Prof. Eremenko I revise the question.

19 years ago, I have heard the following problem from a specialist of dynamical system. During these 19 years, I was in contact with him and I asked him if there is a solution to this problem He always answered me "No, I did not find any answer yet". Now I quote the problem here:

**Problem:**

Is there a real analytic vector field locally defined around origin in $\mathbb{R}^3$, which has an isolated singularity at origin and satisfies the following:

There exist an orbit $\gamma(t)$ which tends to origin as $t$ goes to $+\infty$ and the closure of$\{ \frac{\gamma(t)}{|\gamma(t)|},\;t\in\mathbb{R}^+\}$ has a nonempty interior in $S^2$