In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system

\begin{align} \frac{\partial}{\partial y}\left(\frac{1}{\beta}\frac{\partial\alpha}{\partial y}\right)&=K\:\alpha\beta\\ \frac{\partial}{\partial x}\left(\frac{1}{\alpha}\frac{\partial\beta}{\partial x}\right)&=-K\:\alpha\beta, \end{align}

with $K$ a fixed, real, positive constant. We typically set Goursat data along the axes consisting of $\alpha(x,0)$ and $\beta(0,y)$, so the first question is

Does there exist a unique solution? (At least on points sufficiently close to the initial lines)

(moreover, there should be no reason to restrict $K$ to be a constant, but the same answer may apply if it's a smooth function.)

Setting aside the initial conditions, is it possible to write down some analytic solutions with $\alpha$ and $\beta$ other than polynomials?

This is a cross-post.

In the context of two dimensional elasticity theory, when considering deformations of flat membranes into spherical caps, one encounters the following hyperbolic system

\begin{align} \frac{\partial}{\partial y}\left(\frac{1}{\beta}\frac{\partial\alpha}{\partial y}\right)&=K\:\alpha\beta\\ \frac{\partial}{\partial x}\left(\frac{1}{\alpha}\frac{\partial\beta}{\partial x}\right)&=-K\:\alpha\beta, \end{align}

with $K$ a fixed, real, positive constant. We typically set Goursat data along the axes consisting of $\alpha(x,0)$ and $\beta(0,y)$, so the first question is

Does there exist a unique solution? (At least on points sufficiently close to the initial lines)

(moreover, there should be no reason to restrict $K$ to be a constant, but the same answer may apply if it's a smooth function.)

Setting aside the initial conditions, is it possible to write down some analytic solutions with $\alpha$ and $\beta$ other than polynomials?