Let $\mathbf{x}:U\subset\mathbb{R}^2\longrightarrow\mathbb{R}^3$ be a regular parametric surface with the following properties:
- conjugate net: $\langle \mathbf{x}_{uv}, \mathbf{x}_u\times\mathbf{x}_v\rangle = 0$,
- planar coordinate curves: $\det\left(\mathbf{x}_u,\mathbf{x}_{uu},\mathbf{x}_{uuu} \right) = 0$ and $\det\left(\mathbf{x}_v,\mathbf{x}_{vv},\mathbf{x}_{vvv} \right) = 0$,
- orthogonal carrier planes: $\langle \mathbf{x}_u\times\mathbf{x}_{uu},\mathbf{x}_v\times\mathbf{x}_{vv}\rangle = 0$.
For simplicity, one would probably also require the nondegeneracy conditions:
$$\mathbf{x}_u\times\mathbf{x}_{uu} \neq 0\quad\quad\quad \mathbf{x}_v\times\mathbf{x}_{vv} \neq 0.$$
Example:
The scale-rotation surfaces: $$ \mathbf{x}(u,v) = \left[ \begin {array}{ccc} \mu\left( v \right) \cos \left( \theta \left( v \right) \right) &-\mu \left( v \right) \sin \left( \theta \left( v \right) \right) &0\\ \mu \left( v \right) \sin \left( \theta \left( v \right) \right) &\mu\left( v \right) \cos \left( \theta \left( v \right) \right) &0 \\ 0&0&1\end {array} \right]\, \left[ \begin{array}{c} f(u)\\ 0\\ g(u) \end{array} \right], $$ where $\mu: \mathbb{R}\longrightarrow \mathbb{R}^+ $ and $f(u) \neq 0 $, fits into this class of surfaces.
My Question:
I would like to determine the parametric surfaces possessing the above properties. My knowledge of PDE is very limited and I would appreciate answers putting me on the right track.
1st Edit (July 2022): Does a generic regular surface in $\mathbb{R}^3$, locally support such a parametrization?