this approach, we study the physics of the theories at the enhanced special points in the moduli.

In eight-dimensional $E_8\times E_8$ heterotic strings on the 2-torus $T^2$, only the gauge groups up to $E_8\times E_8\times U(1)^4$ can arise in the perturbative description [@Kimura1810]. This implies that the heterotic dual of F-theory on an extremal elliptic K3 surface with the non-Abelian gauge group of rank 18 does not allow for the perturbative interpretation of the gauge group. This can reflect some non-perturbative aspects of the non-geometric heterotic strings.

F-theory on extremal K3 surfaces and non-geometric heterotic duals in the moduli {#ssec3.2}
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We discuss the points in the moduli of non-geometric heterotic strings with unbroken $\mathfrak{e}_7\mathfrak{e}_7$ algebra, at which the non-Abelian gauge symmetries on the F-theory side are enhanced to rank 18. K3 surfaces as compactification spaces on the F-theory side become extremal at these points. There are finitely many such points in the moduli, and the complex structures and the singularity types of the extremal K3 surfaces that appear in the moduli can be determined from Table 2 of [@SZ]. Among these, those of the singularity types, which include $E_8 E_7$, are studied in [@Kimura1810]. We do not discuss these 