Vector bundles on $\mathbf P^1_\mathbb Z$ with generic fiber $\mathcal{O\oplus O}(1)$ and simple jumps

We study vector bundles on the projective line over a Dedekind domain $A$. In the case where $A$ is a PID, we get a complete classification of rank 2 vector bundles with generic fiber $\mathcal{O\oplus O}(1)$ and with special fibers isomorphic either to $\mathcal{O\oplus O}(1)$ or $\mathcal O(-1)\oplus\mathcal O(2)$.