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I have a White Outdoor LT-1850 Lawn Tractor (MTD). After jumping the
mower because the battery was dead the engine will now not shut off
even if you use the switch. To shut it down you have to disconnect the
battery. After this you can connect the cables back to the battery and
the engine will start up, once it starts up there is a loud buzzing
noise and another thing is the PTO loses function. My first idea is the
starter soleniod is bad. Any ideas?

Just let it run and rent it to the neighbors.

If you connect the battery and the engine starts, I'd say the starter is
staying engaged or it would not turn over by itself. That may be the loud
buzzing noise.