Consider the bundle $\mathcal{O}_{\mathbb{CP}^1}\left(1\right)^{\oplus 2} \to \mathbb{CP}^1$. The collection $A:=\left\{ \left( \bar{b}- \bar{a}\zeta, a +b\zeta \right) \mid (a,b) \in \mathbb{C}^2 \right\}$ of global sections provides a fibration $\phi\colon \mathcal{O}_{\mathbb{CP}^1}\left(1\right)^{\oplus 2} \to \mathbb{C^2}$. That is, through each point of $\mathcal{O}_{\mathbb{CP}^1}\left(1\right)^{\oplus 2}$ there exists precisely one section in $A$ that passes through that point.
It is however clear that this fibration $\phi$ does not descend to a fibration $\mathbb{P}\left( \mathcal{O}_{\mathbb{CP}^1}\left(1\right)^{\oplus 2}\right) \to \mathbb{CP}^1$ since $\lambda s \in A$ corresponds to the section $\left(\bar{\lambda}s_1, \lambda s_2\right)$ and not the section $\lambda\left(s_1,s_2\right)$.
Is there a fibration of $\mathcal{O}_{\mathbb{CP}^1}\left(1\right)^{\oplus 2}$ in this fashion that desends to $\mathbb{P}\left( \mathcal{O}_{\mathbb{CP}^1}\left(1\right)^{\oplus 2}\right)?$
More generally, I am interested whether $\mathbb{P}\left( \mathcal{O}_{\mathbb{CP}^1}\left(1\right)^{\oplus n}\right)$ is holomorphically fibered over its fiber $\mathbb{CP}^n$.