I will be more than happy to help you out with this. Can I get the model number your fridge?
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I do not see a factory installed filter on your machine. Some GE models do not have filters built into them.
There are a couple things that will stop the icemaker from working. If the freezer temperature is too high it will not work. Check the flap where the ice comes through the door. Make sure it is completely closed. If even the slightest bit not fully shut, this will cause the icemaker not to work. Make sure your lights are turning off when the doors are closed. The lights will create too much heat and the freezer will not be cold enough to run the ice cycle. Also, make sure the tube that fills the icemaker on the inside of the freezer is not froze solid. I would run a hairdryer on the tube for about ten minutes to be sure. Ice could form in the bend in the back and may not be visible from inside the freezer.
If this all looks good, then I would investigate a possible defective water valve.
Let me know if you have anymore questions about this. I will be more than happy to help you out further.
Sorry about the delay. Had to step out for a bit. As far as how long before it could freeze again, depends on some variables. It could take years or weeks. Depending on the humidity and average temperature of the room. Worn door gaskets will speed up the process.
It just depends, I tell my customers to think of it as a routine maintainance. Like cleaning the condensor coils on the back, cleaning the door gaskets, changing the filter(if there was one)
If you have anymore questions please ask. I will be more than happy to answer any other questions you may have.