ag.algebraic geometry - Extension of a holomorphic vector bundle

Let $E$ be a holomorphic vector bundle over $mathbb{P}^nsetminusbegin{Bmatrix}[1,0,0,cdots,0]end{Bmatrix}$. 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?