Let $\Phi$ be a $C^{\infty }{ℝ}^n$-action on an orientable $(n+1)$-dimensional manifold. Assume $\Phi$ has an isolated compact orbit $T$ and let $W$ be a small tubular neighborhood of it. By a $C^{\infty }$ change of variables, we can write $W={ℝ}^n/{\mathbb{Z}}^n\times I$ and $T={𝕋}^n\times \left[0\right]$, where $I$ is some interval containing 0.In this work, we show that by a $C^0$ change of variables, $C^{\infty }$ outside $T$, we can make ${\Phi }_{|W}$ invariant by transformations of the type $(x,z)\to (x+a,z),a\in {ℝ}^n$, where $x\in {ℝ}^n/{\mathbb{Z}}^n$ and $z\in I$. As a corollary one cas describe completely the dynamics of $\Phi$ in $W$.