  Atiyah classifies vector bundles on elliptic curves $E$ over an algebraically
closed field of any characteristic. On the other hand, a rank $2$ vector bundle
on $E$ defines a surface $S$ with a $\mathbb{P}^1$-bundle structure on $E$. We
study when $S$ has an elliptic fibration according to the Atiyah's
classification, and what kinds of singular fibers appear.
