If we have the following metric given in $xy$-"coordinates",
$$ \begin{equation} \mathrm{d}s^2=g_{11}(y)\:\mathrm{d}x^2+g_{22}(y)\:\mathrm{d}y^2+2g_{12}(y)\:\mathrm{d}x\:\mathrm{d}y, \end{equation} $$
is there a generic change of variables, that is, some transformation $x=x(u,v),\:y=y(u,v)$, than renders the metric $\mathrm{d}s^2=\alpha^2\: \mathrm{d}u^2 +\beta^2\: \mathrm{d}u^2 $ ?
Of course, every two-dimensional metric can be put in isothermal form, but I want to avoid this argument because it forces me to solve the Beltrami equation, which in general cannot be done analytically. What I wonder is if just from the fact that the metric only depends on one coordinate we can easily guess some change of variables that may diagonalize it.