I read the statement of the Newlander-Nirenberg theorem, which says that "any integrable almost complex structure is induced by a complex structure".
To make sense of the statement, I was wondering how many almost complex structures there are in the first place, i.e. if there is any way to determine all almost complex structures on a given manifold. The sphere admits a complex structure as $\mathbb P^1$, but I know that the complex structure on $\mathbb P^1$ is rigid, i.e. cannot be deformed.
As a next non-trivial example I considered $\mathcal O(k)$, which in real terms would be a twisted rank 2 bundle over the sphere.
My question is now, how can I determine all almost complex structures on $\mathcal O(k)$?