extremal K3 surfaces in this note. Instead, we discuss the extremal K3 surfaces that belong to the moduli, the singularity types of which include $E_7^2$ [^11].

The singularity types of the extremal K3 surfaces in the moduli of K3 surfaces with $H\oplus E_7 \oplus E_7$ lattice polarization, which do not include $E_8$, are as follows [@SZ]: $$E_7^2 A_3 A_1, \hspace{2mm} E_7^2D_4, E_7^2 A_4, \hspace{2mm} E_7^2 A_2^2.$$ We study F-theory on the extremal K3 surfaces possessing the first two singularity types in this note.

Because the perturbative eight-dimensional heterotic strings on $T^2$ can have up to $E_8\times E_8\times U(1)^4$ gauge group, the heterotic duals of F-theory on these extremal K3 surfaces do not allow for the perturbative interpretation of these gauge groups [@Kimura1810]. As we demonstrate in Sections \[sssec3.2.1\] and \[sssec3.2.2\], these theories can be seen as deformations of the stable degenerations as a result of the coincident 7-branes on the F-theory side. These theories satisfy multiple 5-brane solutions on the heterotic side.

### Extremal K3 surface with $E_7^2 A_3 A_1$ singularity {#sssec3.2.1}

The complex structure of the extremal K3 surface with $E_7^2 A_3 A_1$ singularity is uniquely determined, and its transcendental lattice has the following intersection form [@SZ]: $$\begin{pmatrix}
4 