nonperiodic orbits of a smooth $\mathbb{R}$-action on $\mathbb{R}^2$ are embedded lines.

Question

Let $\mathbb{R} \curvearrowright U$ be a smooth action on an open subset $U \subset \mathbb{R}^2$. Show that any non-periodic orbits under the action are embedded lines in U.

My ideas so far:
I have been able to show (using Zorn's Lemma) that for any compact $C \subset U$ that is invariant under the action, there is a minimal, nonempty compact set $C_1 \subset C$ that is invariant.
A hint was given to use Jordan's curve theorem. But I didn't get further.

Any help would be appreciated.