Let $f: X \to Y$ be a regular map of smooth complex irreducible algebraic varieties such that every fiber of $f$ is isomorphic to $\mathbb{C}^{*}$ (the affine line with one puncture). Assume that $(X,f,Y)$ is a locally trivial (wrt Zariski topology) fiber bundle.

Is it true that there exists a rank two vector bundle $E$ on $Y$ such that $X$ is biregular to the complement of two sections of $E$ in the projectivization of $E$ (may be, under certain additional assumptions like the fiberwise $\mathbb{C}^{*}$-action on $X$)? Any relevant references would be greatly appreciated.

EditOops, that is not Zariski locally trivial. Let me keep thinking about this. $\endgroup$