Canonical extensions of ruled surfaces over elliptic curves

Let $L$ be a line bundle of negative degree on an elliptic curve $E$. While the ruled surface $M:=\mathbb{P}({\mathcal{O}}_E\oplus L)$ is one of the simplest varieties in algebraic geometry, its tangent bundle $T_M$ is surprisingly complex: we first show that the ring of symmetric tensors is not finitely generated. We then construct a holomorphic function on the canonical extension $Z_M$ which allows us to show that $Z_M$ is not Stein.