A subbundle of a direct sum of two nonisomorphic line bundles

Question

Let $0 \to N \to L_1 \oplus L_2 \to O_x \to 0$ be a short exact sequence, where $L_1, L_2$ are nonisomorphic line bundles on a smooth projective algebraic curve $X$ and $x$ is a point of $X$. Is it true that $N$ is isomorphic to either $L_1 \oplus Ker(L_2 \to O_x)$ or $Ker(L_1 \to O_x) \oplus L_2$?

Answer 1

A simple example would be to take the non-split exact sequence $0\to O\to E\to O\to 0$ on an elliptic curve $X$. For any point $x\in X$, we have an inclusion $O(-x)\to O$ and when you pull back using this inclusion, one gets an exact sequence, $0\to O\to F\to O(-x)\to 0$. Easy to check that this splits (since $H^1(X, O(x))=0)$ and thus $F=O\oplus O(-x)$ and we have an exact sequence $0\to F\to E\to O_x\to 0$. Dualize to get the exact sequence you want.