I need to show that there is no vector field with transitive orbit on $\mathbb{R}^{2}$.
Let $X:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}$ a $C^{k}, k\geq 1$, vector field in $\mathbb{R}^{2}$. The integral curves of $X$ are the solutions $\phi_{p}(t)$ of the system $x'=X(x)$ through the point $p\in\mathbb{R}^{2}$ defined on its maximal interval $I_{p}$
The orbits of $X$ are the sets $\gamma_{p}=\{\phi_{p}(t):t\in I_{p}\}$.
A transitive orbit of $X$ is a dense orbit of $X$, i.e., $\bar{\gamma_{p}}=\mathbb{R}^{2}$.
Is there some theorem that I can use to solve my question? I tried to suppose that $X$ has dense orbit and get a contradiction, but I didn't got anything.