Stabilizer of the ${\mathbb{R}}^n$-action on a compact connected regular fibre is a full lattice

By [F3], each of the $n$ smooth vector fields restricted to compact $N$ is complete. Their local flows commute by [F2], so their composites define the required global ${\mathbb{R}}^n$-action. For each $p\in N$, [F1] says the orbit map $a\mapsto a\cdot p$ has invertible derivative at zero, so its orbit is open. All orbits are open and partition connected $N$, hence there is one orbit. The same derivative makes the stabilizer $\mathrm{\Gamma }$ discrete, and the orbit map descends to a diffeomorphism ${\mathbb{R}}^n\mathrm{/}\mathrm{\Gamma }\to N$.

Let $W={\mathrm{span}}_{\mathbb{R}}\mathrm{\Gamma }$. If $W\mathrm{\ne }{\mathbb{R}}^n$, the quotient ${\mathbb{R}}^n\mathrm{/}\mathrm{\Gamma }$ maps continuously and surjectively onto the noncompact vector space ${\mathbb{R}}^n\mathrm{/}W$, contradicting compactness of $N$. Thus $\mathrm{\Gamma }$ spans ${\mathbb{R}}^n$.

Choose $n$ real-linearly independent elements of $\mathrm{\Gamma }$, possible by step 2.1, and let ${\mathrm{\Gamma }}_0$ be their integer span. Every coset of ${\mathrm{\Gamma }}_0$ has a representative in their compact fundamental parallelepiped. Since $\mathrm{\Gamma }$ is a subgroup discrete at zero, some $\epsilon$-ball about zero meets it only at zero; translating shows that distinct elements of $\mathrm{\Gamma }$ are uniformly $\epsilon$-separated. Total boundedness of the parallelepiped therefore makes its intersection with $\mathrm{\Gamma }$ finite. Hence $\mathrm{\Gamma }\mathrm{/}{\mathrm{\Gamma }}_0$ is finite and $\mathrm{\Gamma }$ is finitely generated. It is torsion-free as a subgroup of ${\mathbb{R}}^n$, so [F4] makes it free abelian. Since it contains ${\mathrm{\Gamma }}_0\cong {\mathbb{Z}}^n$ with finite index, its rank is $n$. A $\mathbb{Z}$-basis of $\mathrm{\Gamma }$ spans the same real vector space as $\mathrm{\Gamma }$, namely ${\mathbb{R}}^n$, and its $n$ members are therefore real-linearly independent. Thus $\mathrm{\Gamma }$ is a full lattice.