Let $X$ be a smooth complex projective curve of genus $g\geq 1$. Let $\xi\in J^1(X)$ be a line bundle on $X$ of degree $1$. Let $W=\Ext^1(\xi^n,\xi^{-1})$ be the space of extensions of $\xi^n$ by $\xi^{-1}$. There is a rational map $D_{\xi}\colon G(n,W)\rightarrow SU_{X}(n+1)$, where $G(n,W)$ is the Grassmannian variety of $n$-linear subspaces of $W$ and $\SU_{X}(n+1)$ is the moduli space of rank $n+1$ semi-stable vector bundles on $X$ with trivial determinant. We prove that if $n=2$, then $D_{\xi}$ is everywhere defined and is injective.

Let $E$ be a stable rank 2 vector bundle on a smooth projective curve $X$ and $V(E)$ be the set of all rank~1 subbundles of $E$ with maximal degree. Here we study the varieties (non-emptyness, irreducibility and dimension) of all rank~2 stable vector bundles, $E$, on $X$ with fixed $\deg(E)$ and $\deg(L)$, $L \in V(E)$ and such that $\card \bigl( V(E) \bigr) \geq 2$ (resp. $\card \bigl( V(E) \bigr) = 2$).

We show that the use of orbifold bundles enables some questions to be reduced to the case of flat bundles. The identification of moduli spaces of certain parabolic bundles over elliptic surfaces is achieved using this method.