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 $D$?


Edit:For the first question, I just proved that $\widetilde{E}$ is a vector bundle if and only if $\widetilde{E}$ is splits.  I am wondering if this result was already known. If so, does any one know any reference on this result?