How to use Belinsky-Zakharov transformation

I know it might be trivial. When using BZ transformation [1] to generate soliton solutions of Einstein’s field equations, one need a seed solution $g_{0}$ which gives $A_{0}$ and $B_{0}$. Taking them into the equations of Lax pair and find out $\psi_{0}$ which reduce to $g_{0}$ while parameter $\lambda \rightarrow 0$, but how one can find out $\psi_{0}$ from $$D_{1}\psi_{0}=\frac{A_{0}}{\lambda-\alpha}\psi_{0}$$ $$D_{2}\psi_{0}=\frac{B_{0}}{\lambda+\alpha}\psi_{0}$$ Is there a formal procedure to deal with them or you just guess it?

1. V. Belinskii and V. Zakharov, Integration of the Einstein Equations by Means of the Inverse Scattering Problem Technique and Construction of Exact Soliton Solutions, Sov. Phys. JETP 48(6) (1978)

2. https://en.wikipedia.org/wiki/Belinski%E2%80%93Zakharov_transform?wprov=sfti1