in 2.2\]).) Although the presence of two $E_7$ singularities in the Weierstrass equation is explicit in the equation (\[first fibration in 2.2\]), as stated in [@CMSE7], the Weierstrass equation (\[first fibration in 2.2\]) does not necessarily extend over the entire parameter space. For this reason, the Weierstrass equation (\[second fibration in 2.2\]) was instead used to construct non-geometric heterotic strings with unbroken $\mathfrak{e}_7\mathfrak{e}_7$ in [@CMSE7].

The K3 surface with the lattice polarization $H\oplus E_7 \oplus E_7$ (\[hypersurface in 2.2\]) also admits another elliptic fibration further, the singular fibers of which include a type $I_8^*$ fiber (or worse) [@CMSE7]. This alternate fibration relates to $SO(32)$ heterotic string. The Weierstrass equation of this fibration is obtained by making the following substitutions into the equation (\[hypersurface in 2.2\]) [@CMSE7]: $$\begin{aligned}
X & = & tx^3 \\ \nonumber
Y & = & \sqrt{2}x^2 y \\ \nonumber
W & = & 2x^3 \\ \nonumber 
Z & = & 2x^2 (-\varepsilon t+\zeta).\end{aligned}$$ The Weierstrass equation is [@CMSE7] : $$y^2 = x^3 + A x^2 + B x,$$ where $$\begin{aligned}
A & = & t^3-3\alpha t-2\beta \\ \nonumber
B & = & (\gamma t-\delta) (\varepsilon t-\zeta).\end{aligned}$$ The discriminant is given by $$\Delta = B^2 \, (A^2 -4B).$$

