Those symptoms sound similar to a problem that's bitten me in the butt once, then tried to do it in a couple OTHER instances (but I was ready for it :laughing: ) - (of course, it could also just be the overly "cheesy" PTO switch taking a break due to heat)
Solenoid switches (and ignition coils) are wound with REALLY fine magnet wire (bare, single strand wire, insulated by only a varnish coating) - one of the downsides of single strand wire is that there's only one strand to break -
I've had solenoids AND ignition coils do this - either thru vibration, repeated operations or possibly minor stress cracks, the coil winding wire manages to break - but since it's wound tightly inside a coil form, there's no place for the two (now separate) ends to go, so they just SIT there; and as often as not, they are STILL TOUCHING EACH OTHER...
When you first start USING whatever device it is, it's COOL and it's making connection. As things warm up, they EXPAND - now those two halves of the tiny wire can "get away from" each other, and no more current flows. When things cool off, it works again
The (hopefully) GOOD news - there may be a way to troubleshoot this in YOUR time frame with no rushing needed - if you can get at BOTH connections to the PTO, and at least ONE of them has a spade lug connection you can DISCONNECT, you can then connect a DMM across the PTO coil - If you DON'T have at least one of the connections unhooked, the DMM may read OTHER things in the circuit which will confuse the results.
With the mower at rest and the DMM hooked across the PTO coil, you should read a few ohms resistance - with the rear wheels up on ramps (or similar) you should be able to see the PTO - if you then heat up the area around the solenoid with either a high power hair dryer or a heat gun, if there's an intermittent connection as I've described your DMM will (eventually) go to infinity (open, OL, etc)
By testing this way, there's no rush to "find it before it goes away" -
Be careful getting the heat gun too close to the deck belt; they can take some heat, but I wouldn't push it.
I've not had this problem with MY RZT50 (yet) but I HAVE come up with solutions for a couple of OTHER PITA's on this machine involving deck and drive belt replacements so if anyone HATES doing those things I'll be glad to elaborate... Steve