Let $E$, $F$ be two complex elliptic curves, and $A=E \times F$. Let us denote by

$\pi_E \colon A \to E, \quad \pi_F \colon A \to F$

the natural projections. For all $p \in F$ let us write $E_p$ instead of $\pi_F^*(p)$.

Now let us fix $p \in F$ and consider the unique indecomposable rank $2$ vector bundle $\mathcal{F}$ on $A$ defined by the extension

$0 \to \mathcal{O}_A \to \mathcal{F} \to \mathcal{O}_A(-E_p) \to 0$

My main question is the following:

What is the restriction of $\mathcal{F}$ to a fibre of type $E_x$, for $x \in F$ general?

Related to this, another question is

What are $\pi_{F*} \mathcal{F}$ and $R^1 \pi_{F*} \mathcal{F}$?

Observe that the restriction of $\mathcal{F}$ to $E_x$ is given by an extension

$0 \to \mathcal{O}_{E_x} \to \mathcal{F}|_{E_x} \to \mathcal{O}_{E_x} \to 0$,

so by Atiyah's classification of vector bundles over an elliptic curve we have that $\mathcal{F}|_{E_x}$ is either $\mathcal{O}_{E_x} \oplus \mathcal{O}_{E_x}$ or the unique non-trivial extension of $\mathcal{O}_{E_x}$ with itself.

But I cannot decide what happens generically.

**Motivation**. I met this problem when I started the investigation of some triple covers $f \colon X \to A$ such that

$f_*\mathcal{O}_X=\mathcal{O}_A \oplus \mathcal{F}^{\vee}$.

It is worth noticing that
the vector bundle $\mathcal{F}$ is the easiest example of indecomposable vector bundle on $A$ which is *not simple* (in fact, $\textrm{End}(\mathcal{F})=\mathbb{C} \oplus \mathbb{C}$).