Engine stalls, all electrical turns off

My son bought a 1984 KZ550LTD as a first motorcycle to learn to ride. Not gonna lie, it has seen better days, but it was running and it was cheap. It was good for a time, but now it's behaving strange. After it's been running for a while and it's good and warm, it stalls suddenly. There's a loss of power, fuel and oil lights come on, engine stops. At this point all electrical stops working. No lights, no ignition, nothing. Battery is measuring 12.65V. After a brief wait, the lights just come back and everything works again. The engine starts up and it can run for a little while before it stalls again and electrical is off.  It seems to me that something is getting hot and disconnecting until it cools down sufficiently to reconnect, but what? Is there a computer on this thing? The oil light doesn't come on before you start the engine, but comes on before it stalls? The fuel light also comes on before it stalls but it has plenty of fuel. Aren't these just simple circuits that can only work if the sensing switch closes? I really need some help here. What is there other than a main fuse that can cut off all electrical power? I've never heard of a self-healing fuse. 

Welcome to KZRider 
You should post in the new Members forum a intro and include some pictures of your bike
Dave

Do you have the Kawasaki service manual? I'm assuming the motorcycle is a 1984 KZ550-F2 Spectre. Listed as an Ltd-Shaft in the diagrams.

The 550-F2 has a fuel level switch in the fuel tank and an oil level switch in the oil pan, and there should be a self-checker that turns on those warning lights whenever the ignition switch is turned ON; the warning lights should stay on for a certain amount of time predetermined by the checker, then turn off unless the fuel/oil levels are low.  I would compare what the warning lights are doing to what the manual says.
note: the oil level switch could be glitchy or getting stuck. of course make sure the oil level is correct.

Make sure the battery connections are good and the battery cables are good.  check the grounds, especially the main ground usually bolted to the engine near the neg. (-) battery terminal.  I would check the fuses, especially the main fuse. maybe a filament has separated and loses contact? I would go ahead & install new.  Also, check the main fuse box wiring connections. Another possibility is the ignition switch, possibly a broken solder connection inside the switch.

Also, valve clearance check. The engine might be quitting because of tight clearances that get tighter when the engine warms up.