Remove the broken egg of course. I don't know why they rejected an egg. It happens. How old is the female? You really shouldn't breed them until they are a year old but your birds may disa agree! Their first attempt may not be successful. Do you want to breed them? If so, you need to be prepared. Do they have a proper nest box and nesting material? Unlike a lot of parrots, Lovebirds build a nest in their box out of paper, grass, feathers and other materials. Make sure you give them a plentiful nutritious diet. The male will feed the female as she sits on the eggs and he will eat a lot!
They will lay 3 to 7 eggs over a week to 10 days and will start full time incubation after a few eggs are laid. After about week of incubation you can candle the eggs with a flashlight to see if they're developing. If they're fertile you will see a network of bright red blood vessels around a red embryo. The eggs should hatch after about 22 to 25 days of full time incubation.
If you're really serious about breeding your Lovebirds there are some things you need to invest in. I recommend you purchase some hand feeding formula like Kaytee Exact or Higgins InTune (I prefer Higgins) in case the parents reject a chick and you need to take over feeding. I also recommend you purchase a thermostatically controlled egg incubator that can double as a chick brooder. I have one I bought on Amazon, Janoel-12, for about $50 and it's perfect for both eggs and chicks just in case the parents reject a chick. You can't keep a naked baby parrot alive using a light bulb or heating pad for warmth so don't even try it. They are very delicate creatures and are unable to move around themselves to find the best temperature like chickens can. Even a few degrees too hot or cold and the hatchlings will die.
Raising baby parrots yourself is a difficult time consuming endeavor but watching a baby parrot die because it's parents reject it is even
worse, so I recommend you be prepared to intervene if necessary. If you have any questions I'd be happy to help. I've successfully hand raised baby budgies from hatching several times.
Good luck!
