Bundles on $\mathbb{P}^1_\mathbb{Z}$ of rank $3$ and non-degenerate sections of bundles of rank $2$

A classification of rank $3$ bundles with a trivial generic fiber and simple jumps is obtained. Using the resulting classification, it is proved that two bundles $E$ and $F$ of rank $2$ with a trivial generic fiber and simple jumps with equal discriminants are stably isomorphic, that is, $E\oplus\mathcal{O}\simeq F\oplus\mathcal{O}$. In the second part of the work it is shown that for a rank $2$ bundle with a trivial generic fiber there are non-degenerate sections of all degrees higher than minimal one.