We consider $TS^{2}$ as a 2 dimensional holomorphic manifold and fix an explicit holomorphic structure on $TS^{2}$ as it is indicated in the answer of Mike Usher to the following question. We have two questions which are not necessarily related to each others(directly).

1)Is there a

holomorphicmap $q:TS^{2}\to S^{2}$ such that $(TS^{2}, S^{2}, q)$ has an structure of a holomorphic line bundle or at least topological one dim. complex line bundle or two dim real vector bundle?2) Consider the natural projection $p:TS^{2}\to S^{2}$. By definition, a holomorphic vector field on $S^{2}$ is a holomorphic map $X:S^{2}\to TS^{2}$ with $p\circ X=Id.$ To what extend these holomorphic vector fields have been classified? In particular what can be said about the nature of singularities and closed orbits of a holomorphic vector field? Can we have a singularity with negative index, ex: saddle point? Can we have a limit cycle for a holomorphic vector field? Is the space of holomorphic vector field a Lie algebra, that is closed under the usual Lie bracket?

**Note 1:** In the above questions, $S^{2}$ is identified with the one dim complex manifold $\mathbb{C}P^{1}$

**Note 2:** The second question is weakly motivated by the following facts: Let $F:U\to \mathbb{C}$ be a holomorphic map where $U\subset \mathbb{C}$ is an open subset. Then the index of each singular point of the vector field $\dot x=F(x)$ is non negative. Moreover such vector fields have no limit cycle, because $[F,iF]=0$ on the other hand every two commuting flows share on limit cycles, that is every limit cycles of one is invariant under the other ones. The other reason that $\dot x= F(x)$ has no limit cycle is that for each $t$ the flow $\phi_{t}$ is a holomorphic map Now put $T=\text{the period of limit cycle}$. So $\phi_{T}$ as a holomorphic maps has a curve of fixed points. this implies that $\phi_{T}$ is identicaly equal to the identity map. So a closed orbit lies in a band of closed orbit with the same period. so there is no an isolated closed orbit, i.e. limit cycle.