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.

You can call Jack’s on the carb, they have more technical info.
If you use carb cleaner or starting fluid in carb does it start backup.

Luc55
When it starts to sputter, I've sprayed starting fluid into the carburetor and it immediately picks back up and runs a couple of minutes longer

Okay, you've just narrowed it down quite a bit.  It's fuel related..
Under load, the engine, generally, needs slightly more fuel, so it may be leaning out
once hot and under load. 
 
A new carb likely won't fix this (un-less yours is clogged, which it doesn't sound like it).
Richening up the mix may help.  Kawasaki may have various jets available for the machine.
If it is narrowed down to jetting, drilling a slightly larger hole in the main jet(s) would provide 
more fuel...  There are very , very small drill bits available for this. 
 
 
"Shocking" the fuel with say "Sea Foam" , if the carb is clogged could help (certainly can't hurt)..
 
Can you post a picture of the spark plug here WHEN IT HAPPENS?
We need to "read" the plug    
 
 
Just for S & G's, try attaching an independent fuel tank and have it gravity feed direct to the carb. 
See if that makes a difference... 
 
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