Construction of a linear filtration for bundles of rank $2$ on $\mathbf{P}^1_{\mathbb Z}$

We obtain an algorithm for the construction of a filtration with linear factors for vector bundles of rank 2 over the surface $\mathbf{P}^1_A$, where $A$ is a Euclidean domain. In other words, we produce an algorithm that, for an invertible $2$-matrix $\sigma$ over the ring $A[x,x^{-1}]$, constructs matrices $\lambda$ over $A[x]$ and $\rho$ over $A[x^{-1}]$ for which $\lambda\sigma\rho$ is an upper triangular matrix.
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