Classifying $GL(2,\mathbb Z) \ltimes \mathbb Z^{2}$-orbits by subgroups of $\mathbb R$

Let $\mathcal G_2$ denote the affine group $GL(2,\mathbb Z) \ltimes \mathbb Z^{2}$. For every point $x=(x_1,x_2) \in \R2$ let $\orb(x)=\{y\in\R2\mid y=γ(x)$ for some $γ\in \mathcal{G}_2 \}$. Let $G_{x}$ be the subgroup of the additive group $\mathbb R$ generated by $x_1,x_2, 1$. If $\rank(G_x)\in \{1,3\}$ then $\orb(x)=\{y\in\R2\mid G_y=G_x\}$. If $\rank(G_x)=2$, knowledge of $G_x$ is not sufficient in general to uniquely recover $\orb(x)$: rather, $G_x$ classifies precisely $\max(1,ϕ(d)/2)$ different orbits, where $d$ is the denominator of the smallest positive nonzero rational in $G_x$ and $ϕ$ is Euler function. To get a complete classification, polyhedral geometry provides an integer $c_x\geq 1$ such that $\orb(y)=\orb(x) $ iff $(G_{x},c_{x})=(G_{y},c_{y})$.

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