Z665 shuts off when mowing

John Deere Z665, shuts off after 15-20 minutes of mowing. Subsequent start attempts result only in the starter solenoid clicking. After 30 minutes or so the mower will start and run for about 15 minutes.

Any ideas?

You first have to work out if it is fuel or spark.
Thus you get a can of starter fluid & neon tube spark tester, fit it to the mower, remove the hood so you can see it then go mow.
When the mower plays up look at the tester .
no red flashes = replace magneto
Red flashes but no go give it some starter fluid.
Fires right up, = fuel supply problem
Most common fuel supply problems
1) blocked vent in cap
2) debris forming a plug in the tank outlet

the starter solenoid does not engage the starter so it cannot be fuel related.

Sorry, my bad, did not read the post properly.
Overheating is the first thoughts.
Pull off the blower housing and check for debris blocking the fins.
Turn the engine over by hand when cold for refference.
When it stops try turning the motor over again.
If it is a lot harder to move then your are seizing.