Vector bundles on $\mathbf P^1_\mathbb Z$ with simple jumps

We consider vector bundles with rank 2 over the projective line over $\mathbb Z$. Assume that such a bundle $E$ is trivial on the generic fiber, and its restriction to any special fiber is isomorphic either to $\mathcal O^2$ or to $\mathcal O(-1)\oplus\mathcal O(1)$. Under these assumptions we prove that there exists an exact sequence of the form $0\to\mathcal O(-2)\to E\to\mathcal O(2)\to0$.