Structures emerging in a discrete plot of palindromic numbers

All points are formed by the polynomials as you said. When the graphs of these polynomial functions are too sparse, secondary structures start to appear. You can generate these structures like this:

Take one of the polynomial and replace the coefficients by functions of x and y. The resulting function should intersect the $\Bbb{N}^2$ grid as densely enough and it should have small first and second derivations. If this is met on an interval, you will see a secondary structure in the given form.

If you start with linear functions of y, you will get most of the structures, mainly the parabolas. You can also try other polynomial functions of x and y.

There are also thick structures which are formed when a few similar structures are located close together. And there are also thick gaps which are formed when there are no structures in the given area.

The question is still: Which functions yield the noticeable structures?

I think, this problem is similar to the research of prime numbers or the Collatz conjecture. There seems to be an order but it's really hard (if not impossible) to predict it.