The irrational torus flow is free with dense orbits

The displayed map is jointly smooth. Its phase factors satisfy $e^{2\pi i(s+t)}=e^{2\pi is}{e}^{2\pi it}$ and likewise with $\alpha (s+t)$, so $0$ acts as the identity and $(s+t)\cdot x=s\cdot (t\cdot x)$; hence it is a smooth left action by [F1]. If $t$ fixes any $(z,w)$, then $e^{2\pi it}=e^{2\pi i\alpha t}=1$, so $t\in \mathbb{Z}$ and $\alpha t\in \mathbb{Z}$. Irrationality forces $t=0$, proving freeness.

The integer rotation orbit $\{n\alpha +\mathbb{Z}:n\in \mathbb{Z}\}$ is dense in $\mathbb{R}\mathrm{/}\mathbb{Z}$. Indeed, given $\epsilon >0$, choose an integer $N>1\mathrm{/}\epsilon$ and partition $[0,1)$ into $N$ half-open intervals of length $1\mathrm{/}N$. Applying [F2] to the $N+1$ fractional parts of $0,\alpha ,\dots ,N\alpha$ gives integers $0\le r<s\le N$ with $\parallel (s-r)\alpha \parallel <1\mathrm{/}N<\epsilon$. Replacing $q=s-r$ by $-q$ if necessary makes $\delta =\{q\alpha \}$ satisfy $0<\delta <\epsilon$. For any $u\in [0,1)$, the integer $m=\lfloor u\mathrm{/}\delta \rfloor$ satisfies $0\le u-m\delta <\delta$ and $m\delta <1$, so the orbit point $mq\alpha +\mathbb{Z}=m\delta +\mathbb{Z}$ is within $\epsilon$ of $u+\mathbb{Z}$.

Fix a source point $(z,w)$ and a target $(e^{2\pi ia}z,e^{2\pi ib}w)$. Parameters $t=a+n$, $n\in \mathbb{Z}$, put the first coordinate exactly at $e^{2\pi ia}z$, while their second-coordinate phases are $e^{2\pi i\alpha a}{e}^{2\pi in\alpha }$. Step 1.2 makes the latter dense in $S^1$, so some such parameters put the action point arbitrarily close to the target. Hence every orbit is dense.

The orbit map $i$ is a homomorphism by the phase calculation in step 1.1. It is injective because $i(t)=(1,1)$ implies $t=0$ by the freeness calculation. Its differential sends $1\in T_0\mathbb{R}$ to the nonzero tangent vector $2\pi i(1,\alpha )$; translating the homomorphism identity shows its differential is injective everywhere. Thus $i$ is an injective immersion. All choices above are finite or single choices, so no choice principle is used.