The [Quillen-Suslin][1] theorem asserts that there are no nontrivial vector bundles over the affine space $\mathbb{A}^{n+1}$, $n\geq 0$.
Let's work over the complex numbers. What can be said about vector bundles on the *punctured* affine space $X_n=\mathbb{A}^{n+1}\smallsetminus\{0\}$?
According to [this paper][2], there seem to be room for nontrivial vector bundles.

Let $\mathbb{C}^{*}$ act on $X_n$ by the action $\lambda.(x_0,\dots,x_n):=(\lambda x_0,\lambda x_1,\dots, \lambda x_n)$ whose quotient is $\mathbb{P}^n$.
Notice that equivariant v.b. on $X_n$ are in bijection -via pullback- with v.b. on $\mathbb{P}^n$, and the latter form already a rich moduli problem on its own. In this question we concentrate on the specificity of $X_n$

> **1.** Is there some sort of classification of v.b. on $X_n$, taking as a starting base -say- the "classification" of stable v.b. on $\mathbb{P}^n$ given by the corresponding moduli spaces?

> What about particular ranks, for example the case of **line bundles**?

>  **2.** Are there vector bundles on $X_n$ that are not pullbacks of v.b. on $\mathbb{P}^n$, that is, v.b. on $X_n$ that do not admit an equivariant structure?











  [1]: https://en.wikipedia.org/wiki/Quillen%E2%80%93Suslin_theorem
  [2]: https://arxiv.org/pdf/1303.0575.pdf