Kawasaki FR691v cuts out and stalls

I have an Ariens Ikon XD 60 zero turn with a Kawasaki FR691v CS30 r (more about that later. When sitting still or not cutting, it will run steadily. When we engage the blades, it may run fine for about 20 minutes then start cutting out and then stalls as if it is out of gas. Looking at various forums, I see where many others have had the same problem but I can't find the fix. I have worked on it then carried it to a repairman twice. Still cuts off while cutting. The coils have been replaced as have the fuel pump, spark plugs, fuel cap, fuel lines, and fuel filter. Still have the same problem. I have considered replacing the carburetor which brings up the next problem. I cannot find replacement parts for the FR691v CS30-r anywhere. Don't even find a mention of that specific model engine anywhere. So I am searching for a solution to the mower's problem and an answer as to why I can't find anything about that particular model.