Let $E=G=\cos^2u$, $F=0$, $\displaystyle e=\frac{1}{\sqrt{\cos 2u}}$, $\displaystyle g=\sqrt{\cos 2u}$ and $f=0$, where $\displaystyle u\in \left(-\frac{\pi}{4}, \frac{\pi}{4}\right)$. Does there exist a regular surface in $\mathbb{R}^3$ with parametrization $\textbf{x}(u, v)$ whose first and second fundamental forms are these six given functions? If yes, find one such parametrization $\textbf{x}$.
I know such a regular surface exists because the six given functions satisfy the Gauss formula and Mainardi-Codazzi equations. For example, the Gauss curvature is $$ K=\frac{eg-f^2}{EG-F^2}=\frac{1}{\cos ^4u}$$ which is also given by the Gauss formula $$-\frac{1}{2\sqrt{EG}}\left(\left(\frac{E_v}{\sqrt{EG}}\right)_v+\left(\frac{G_u}{\sqrt{EG}}\right)_u\right)=-\frac{1}{2\cos ^ 2u}\cdot\frac{d}{du}\frac{-2\cos u\sin u}{\cos^2u}=\frac{1}{\cos^4u}.$$ How can we find a parametrization $\textbf{x}$ of such a regular surface?