How to go from the wave equations of electric and magnetic field and $$ \boldsymbol{\nabla}\cdot \mathbf E = 0 \quad \text{ and } \quad \ 0 = \boldsymbol{\nabla}\cdot\mathbf B, $$
to the remaining two Maxwell's equation in free space?
I am unable to do this, for a long time now. Please help. :) Is it even possible to do it?
The question arose from the following: Are Wave equations equivalent to Maxwell equations in free space?