Exercise 29, § 2

Let $G$ be the topological group $\mathbf{R}$, let $\varphi$ be the canonical mapping of $\mathbf{R}$ onto its quotient group $T = \mathbf{R}/\mathbf{Z}$, and let $\theta$ be an irrational number. The group $G$ operates continuously on $E = T^2$ by the rule $(s, (x, y)) \to (x + \varphi(s), y + \varphi(\theta s))$. Show that the stabilizer of each $z \in E$ consists of $0$ alone, that the orbit of $z$ is dense in $E$ and that it is not a topological homogeneous space relative to $G$.*