elliptic K3 surface are enhanced to rank 18, K3 surfaces become [*extremal*]{} K3 surfaces. A K3 surface is called attractive, when it has the Picard number $\rho=20$, which is the highest value for a complex K3 surface. When an attractive K3 surface has an elliptic fibration with a section with the singularity type of rank 18, the elliptic fibration is referred to as extremal. Owing to the classification result in [@SZ], the complex structures of extremal K3 surfaces on which non-Abelian gauge groups on F-theory compactifications are enhanced to rank 18 in the moduli can be determined, and this enables us to deduce the Weierstrass equations of extremal K3 surfaces. By analyzing the deduced Weierstrass equations, we study F-theory compactifications and the non-geometric heterotic duals at these special points in the moduli.

We also discuss applications to $SO(32)$ heterotic strings in this study. We deduce the Weierstrass equations of elliptic K3 surfaces appearing as the compactification spaces of the F-theory duals of some $SO(32)$ heterotic strings, which are obtained as the transformations of $\mathfrak{e}_7\mathfrak{e}_7$ non-geometric heterotic strings.

In addition, we consider fibering elliptic K3 surfaces that belong to the F-theory side of the moduli of eight-dimensional $\mathfrak{e}_7\mathfrak{e}_7$ non-geometric heterotic strings, 