Let $E$ be a holomorphic vector bundle over $\mathbb{P}^n\setminus\begin{Bmatrix}[1,0,0,\cdots,0]\end{Bmatrix}$. Let $D$ be a connection on $E$. Let $\widetilde{E}$ be an extension of $E$. Since $\widetilde{E}$ is reflexive, i.e. double dual of $E$ is isomorphic to itself, then up to isomorphism, $\widetilde{E}$ is unique. My questions are

1) Is it possible that $\widetilde{E}$ is a vector bundle?

2. If $\widetilde{E}$ is a vector bundle, does it admit a connection $\widetilde{D}$ which is naturally induced by $D$?