# Spaces of sections of a holomorphic fiber bundle with specific normal bundles

There is a well-known fact (see for example HKLR) that if $$p\colon Z\to \mathbb CP^1$$ is a holomorphic fiber bundle admitting a holomorphic section $$s\colon \mathbb CP^1\to Z$$ such that $$s^*N\cong \mathcal O(1)^{\oplus 2n}$$ where $$N$$ is the normal bundle then the space $$M$$ of sections with such a normal bundle is of complex dimension $$4n$$ and admits a biquaternionic structure i.e. the action of $$Mat_2(\mathbb C)$$ on $$TM$$. If $$Z$$ is eqipped with an antiholomorphic involution $$\tau$$ restricting to the standard antiholomorphic involution on $$\mathbb CP^1$$ when the space $$M^\tau$$ of $$\tau$$-invartiant sections admits a natural hypercomplex structure. In this case one can reconstruct $$Z$$ from $$M^\tau$$ as its twistor space.

I'll remind how one gets the biquaternionic structure. Kodaira-Spencer tells us that $$T_sM\cong H^0(s,N)$$. There is also a vector bundle $$E$$ on $$M$$ such that $$E_s\cong H^0(s,N(-1))$$. We have a natural morphism $$\alpha\colon E\otimes \mathbb C^2\to TM$$ which will be an isomorphism if and only if $$N$$ is a direct sum of $$\mathcal O(1)$$. The tautological action of $$Mat_2(\mathbb C)$$ on $$\mathbb C^2$$ gives us an action on $$TM$$.

I've constructed a holomorphic fiber bundle $$p\colon Z\to \mathbb CP^1$$ with a $$4n$$-dimensional family of holomorphic sections with normal bundle $$\mathcal O^{\oplus n}\oplus \mathcal O(2)^{\oplus n}$$. Do spaces of sections of such fiber bundles admit any interesting structures? I understand that the subbundle of $$TM$$ corresponding to $$H^0(\mathcal O(2)^{\oplus n})$$ admits a natural action of $$Mat_3(\mathbb C)$$. And even more interesting question

Is there some way to reconstruct the given fiber bundle from the space of its holomorphic sections if the sections with normal bundle $$\mathcal O(1)^{\oplus 2n}$$ do not exist?

By the way, I don't think that there exist an antiholomorphic involution on my fiber bundle (only with essential singularities on $$p^{-1}(\{0,\infty\})$$) but there exists a holomorphic one.