Supose we have a charge $+q$ which is held at a distance $d$ from the plane $z=0$ not grounded. Very similar to the classic case: Electric induction in a grounded plane conductor

But the infinite plane is not grounded. Which would be the charge distribution in the plane then?

I assume that the charge conservation will give it null total charge. But unlike the sphere, which is a closed surface, we can't just add $\sigma = + \frac qA$ because that would just be zero.

How can we solve Laplace's equation for this case? Should we solve it for a finite plane and take the limit?