I'm sure I can do that with array and some distortion, but how? I use Adobe Illustrator, and I've only tried the pen tool to draw each circle by hand so far, which shouldn't be the smartest thing to do. By the way, I need the exact same form.
A guess: You have it only as a low resolution bitmap and you need a vector version. In this case trace it and forget the idea to make an exact replica by applying some distortion transform to a less curved pattern of ellipses. There's too many unknown variables that should be guessed right. A mathematician might find the wanted transformation rule, but for most of us it's undecidable (=can be found only by an accident).
If you have it as vector and only want the transformation to make a replica starting from ellipses I cannot help.
Here's a snippet of the tracing result layered on your image and colored to blue.
This it the whole result(black is your image):
I'm sure it's as good as it's possible to get without any special software which tries to fit certain limited set of curve forms to your bitmap image.
I traced it in Inkscape because its default settings work well. In addition Inkscape's Node tool is extremely well designed to make fixes to paths. For example if you remove a node to get rid of an unwanted twist the curve stays as close as possible the original form, only smoother. In Illustrator removing a node generates a shortcut.
The original array may have had rotated ellipses (at the left had side). Possibly using a step blend to make each row. Maybe something like this
Then distorted with a 2 x 2 Envelope Mesh
This is fine if you are happy enough with something similar, however since you want the exact same form, I think it would be really difficult to use that method to recreate it exactly. Also redrawing and distorting circles manually is possible, but a lengthy process.
You could instead do an Image Trace. You can see the settings I used here.
The result of this depends on the quality of the original, which isn't great to be honest. However, if necessary, you could expand the trace and fix some of the curves manually if they got messed up.