We have heard ( from forum comments ) a number of complaints recently that the chemicals used in water treatment facilities are causing the material the water reservoirs are made from to break down and particles appearing as white flakes in the dispensed water. In that case, replacing the reservoir is usually necessary.

Most manufacturers originally used a molded plastic water reservoir which is what seems to be breaking down. Many fridge manufacturers are now selling a coil of plastic tubing (like used in the rest of the water system) in place of the molded plastic reservoir. You can see some of the replacement kits at the following link:

Whirlpool fridge water reservoir replacement kit

Frigidaire water reservoir replacement kit

Maytag water reservoir tube kit

It is unknown at the moment if GE has gone to one of these tubing kits but you can see an original molded plastic water reservoir that is used on GE refrigerators at the following link

General Electric fridge water reservoir

Where is my water reservoir? Usually behind the crispers in the fresh food section....more help below if needed. 

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Related links - Refrigerator SxS -  Interactive  

Related links -  How to change a common water tank/reservoir