Let $ \Omega_s(x) $ solve the system

$$s \dfrac{\partial^2}{\partial s^2}\Omega_s(x)=\pm x\dfrac{\partial}{\partial x}\Omega_s(x) $$

$$2\sqrt{s}\frac{\partial}{\partial s} \sqrt{\pm\Omega_s(x)}=\sqrt{x} ~\Psi \Omega.$$

where

$$\Psi \Omega := \sqrt{\mp\frac{\partial}{\partial x}{\Omega_s(x)}}$$

on some domain $D\subset\Bbb R^2_{\gt 0}$, with Cauchy data $\Omega_s(x)|_{s=0}.$ Assume $\Omega$ is real analytic, thus it has some holomorphic extension, $\tilde \Omega.$

By transforming the system via $s\mapsto is$ we see that $\Delta(s,x)=\tilde \Omega(is,x)$ solves

$$is \dfrac{\partial^2}{\partial s^2}\Delta(s,x)=\mp x\dfrac{\partial}{\partial x}\Delta(s,x) $$

and we can write down the transformed (nonlinear) equation in a similar manner.

I'm interested in the existence of solutions and any sort of structure the solutions may have, as well as uniqueness for the original and transformed system.

What can be said?

There may only be certain solutions with no nice structure involved do to the nonlinearity of the system but I welcome any solutions at all.