Fix a hyperkähler manifold $X$ and an identification of $S^2$ with the hyperkähler sphere of $X$. Now consider the twistor space $T := S^2\times X$ equipped with the tautological complex structure. For each $x\in X$, we have a holomorphic map $u_x:S^2\to T$ defined by $u_x(\theta):=(\theta,x)$.

**Question:** Is every holomorphic map $u:S^2 \to T$ which satisfies ${\rm pr}_1\circ u={\rm id_{S^2}}$ of this form?