coordinates on $\P^3$. $\alpha, \beta, \gamma, \delta, \varepsilon, \zeta$ are parameters, and $(\gamma, \delta)\ne (0,0)$, and $(\varepsilon, \zeta)\ne (0,0)$.

Making the following substitutions $$\begin{aligned}
X & = & tx \\ \nonumber
Y & = & y \\ \nonumber
W & = & 4t^3 \\ \nonumber 
Z & = & 4t^4\end{aligned}$$ yields the Weierstrass equation with two type $III^*$ fibers as follows [@CMSE7] : $$\label{first fibration in 2.2}
y^2 = x^3 + 4t^3\, (\gamma t^2 -3\alpha t+\varepsilon)\, x-8t^5\, (\delta t^2+2\beta t+\zeta).$$ Type $III^*$ fibers are at $t=0$ and at $t=\infty$.

K3 surface with the lattice polarization $H\oplus E_7 \oplus E_7$ given by (\[hypersurface in 2.2\]) always admits another fibration with a type $II^*$ fiber and a type $I^*_2$ fiber, as shown in [@CMSE7], and the Weierstrass equation of this fibration is: $$\label{second fibration in 2.2}
\begin{split}
y^2 = & x^3 - \frac{1}{3} \big[9\alpha t^4+3(\gamma\zeta +\delta\varepsilon)\, t^3+ (\gamma\varepsilon)^2\, t^2 \big] x\\
 & + \frac{1}{27} \big[27t^7 - 54\beta t^6 + 27 (\alpha\gamma\varepsilon+\delta\zeta)\, t^5+9\gamma\varepsilon (\delta\varepsilon +\gamma\zeta)\, t^4+2 (\gamma\varepsilon)^3\, t^3 \big].
 \end{split}$$ The Weierstrass equation (\[second fibration in 2.2\]) was used in [@CMSE7] to construct eight-dimensional non-geometric heterotic strings with unbroken $\mathfrak{e}_7\mathfrak{e}_7$. (Compare the equation (\[second fibration in 2.2\]) with the equation (\[Weierstrass with param 