Rank $2$ spanned vector bundles on $\mathbb {P}^2$ with a fixed restriction to a line or a prescribed order of stability

Fix a line $D\subset \mathbb {P}^2$. In this note we study rank $2$ spanned vector bundles with prescribed Chern classes and either with a prescribed order of stability or whose restriction to $D$ has a prescribed splitting type, mainly when the splitting type is either rigid or the most extremal one, $(c,0)$.We use the description of the Chern classes of all rank $2$ spanned bundles due to Ph. Ellia.