Things like ignition coils, solenoid coils, etc, are almost always wound with solid (single strand) wire (called magnet wire) - the only insulation is usually varnish, so more turns of wire (more magnetic effect) can be put on the same size spool.
Because the internal wire is only one strand (unlike the more flexible, stranded hook-up wire used everywhere else) if that strand breaks, you're DONE.
The frustrating part - often you will get a single break INSIDE a sealed coil unit (like your PTO clutch solenoid) - when the coil is COLD, the two (broken) ends still touch, and the clutch will work. Warm it up by running current through it and it expands, causing the broken halves to separate - it'll shut off. Cool it down a while, things contract, it works again.
Repeat til insanity sets in
If it's possible to QUICKLY get at the connections to your clutch coil, (usually they will have push-on spade connectors) first check continuity of the coil by removing one lead and measuring across the coil with an ohm meter - should read less than 20 ohms or so - then hook it back up, start the machine, engage PTO, let it run til it stops the blade, then -
Shut off the motor and QUICKLY pull the same wire off (so you're NOT reading other things in the circuit) and check resistance across the coil - should be OPEN (infinite) - keep watching the meter til the coil cools - if I'm right, your reading should return to what you got before starting the mower.
If you're lucky, all that's left is to get the part# and order a new coil... Steve