Consider a flat, simply connected surface endowed with the Riemannian metric $g_0=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +\mathbb{d}^2v \right)$, so that $\Omega(u,v)$ is an arbitrary harmonic function. Now we "shift" the metric to $g=e^{2\Omega(u,v)}\left(\mathbb{d}^2u +2\cos\gamma\:\mathbb{d}u\:\mathbb{d}v+\mathbb{d}^2v \right)$ and wish to determine all the functions $\gamma=\gamma(u,v)$ that leave the surface flat. It follows that such a function satisfies
\begin{align} \frac{1}{\sin\gamma}\frac{\partial^2 \gamma}{\partial u \partial v}+\boldsymbol{\Delta}_\gamma\Omega=0 \tag{*}\label{*}, \end{align}
with the operator on the l.h.s being the Laplace-Beltrami operator associated with the metric $g_\gamma=\mathbb{d}^2u +2\cos\gamma\:\mathbb{d}u\:\mathbb{d}v+\mathbb{d}^2v$, and given by
\begin{align} \boldsymbol{\Delta}_\gamma=\frac{1}{\sin\gamma}\left[ \frac{\partial}{\partial u}\left(\csc\gamma\frac{\partial\:\:}{\partial u} \right)- \frac{\partial}{\partial u}\left(\cot\gamma\frac{\partial \:\:}{\partial v} \right)+(u\longleftrightarrow v)\right].\tag{**} \end{align}
(we further assume that $0<\gamma<\pi$.) The equation (\ref{*}) is clearly hyperbolic, but it is not entirely clear what are the existence and uniqueness conditions of its solutions, and if some non-trivial solutions may be obtained from more generic diffeo-geometric arguments.
One particular case of interest is $\Omega(z=u+iv)=\Omega_0-\log\left|1-\bar{a}z\right|^2$, with $a$ and $z$ in the unit disk, which is the family of functions associated with conformal automorphisms of the unit disk.
(This equation is encountered in the study of woven sheet fabrics, and some other applications of Chebyshev networks to structural mechanics.)
