  Let $X:=\mathbb{A}^{n}_{R}$ be the $n$-dimensional affine space over a
discrete valuation ring $R$ with fraction field $K$. We prove that any pointed
torsor $Y$ over $\mathbb{A}^{n}_{K}$ under the action of an affine finite type
group scheme can be extended to a torsor over $\mathbb{A}^{n}_{R}$ possibly
after pulling $Y$ back over an automorphism of $\mathbb{A}^{n}_{K}$. The proof
is effective. Other cases, including $X=\alpha_{p,R}$, will also be discussed.
