We have a female Umbrella Cockatoo, age 26 who, despite head-only petting when she is 'broody' will lay an occasional egg or two (maybe once a year, sometimes not for a couple years).
I have never been able to find the answer to: Is it better to let her sit on the eggs or take them away immediately? We have tried both methods. She gets a very healthy diet and supplemental calcium when she is going to lay.
I have read that if you take away the eggs, it encourages them to lay more, but I worry that sitting on the eggs for several weeks isn't good for her either. Has any research been done on this topic?
The amount of eggs your bird is producing is not a problem, but it is best to let her keep them so she does not try to lay another in its place over and over. If they crack, remove them. You can also replace the eggs with a ball of similar size.
Do you have a nest or box or cozzy she sleeps in? Take it out. Give her a shallow, open, not dark box for her eggs if she has them now, then discard it all when she is done.
I recommend rearranging her cage so it is all new to her and that will help distract her from being too comfy and wanting to raise a family.
Overall, our handling our birds like children, stroking and petting them only serves to arouse them and start their urge to lay. We have to restrict our petting to their heads.
Is she also getting 10-12 hours of undisturbed darkness each night? A cage cover or a light blanket is a good start. Or move her to a sleeping cage at night in a quiet, dark room and bring her back out during the day.
Strange (to your bird) light cycle can trigger egg laying. Full spectrum lighting on a timer near your birds cage can help correct this
I hope this helps.