Kindly refer to "The classic image problem" of "THE METHOD OF IMAGES" of Introduction to electrodynamics book by Griffiths (Chapter 3).
A point charge is placed at $(0,0,d)$ on the $z$ axis. The $XY$ plane is a infinite CONDUCTING plane which is grounded. The author calculated the potential by the method of images. Then the author also calculates the induced charge on the conducting plane. The charge distribution comes out to be
This suggests that the accumulation of charge is highest at $x=y=0$ and it gradually decreases as the value of $x$ and $y$ increases.
My question is as follows.
The plane is a conducting plane. How can it support a non uniform distribution of charge? Any non uniformity in charge accumulation is supposed to be distributed evenly since the plane conducts.