Let $f:Y\rightarrow\mathbb{P}^1$ be an elliptic $K3$ surface, then the holomorphic 2-form $\Omega_Y$ vanishes when restricted to an elliptic fiber $f^{-1}(b)$ with $b\in\mathbb{P}^1$. After a hyperkahler rotation, we get a new $K3$ surface $X$, and the map $f:X\rightarrow S^2$ becomes a special Lagrangian fibration with respect to the Calabi-Yau structure $(\omega_X,\Omega_X)$ on $X$. For $X$, one can then talk about its SYZ mirror $X^\vee$. Note that we have an $S^1$ family of choice of such $X$ so that $f:X\rightarrow S^2$ is a special Lagrangian fibration. By choosing the symplectic structure $\omega_X$ on $X$ appropriately, the mirror symmetry of $X$ can be interpreted by a hyperkahler rotation. Applying hyperkahler rotation we get the mirror $X^\vee$, which is still a $K3$ surface, the underlying differentiable structure is unchanged but with a different complex structure. We can choose $\omega_X$ so that $f:X^\vee\rightarrow S^2$ is still a special Lagrangian fibration with respect to $\omega_{X^\vee}$ and $\Omega_{X^\vee}$.
Note that in general $X^\vee$ may not be an elliptic $K3$ surface. My question is when will $X^\vee$ be an elliptic $K3$?
