Yes the hst can cause your engine to overheat. There's an oil cooler at the filter. If your oil is excessive hot it can overcome the engine cooling. Get a laser heat gun and check the different locations for heat
Zetor made in Czech Republic. Belarus are made in minsk belarus but they sold a bigger 4wd made in Ukraine and another even bigger from Russia. All still available and parts.
Check the oil level in the gearbox? Has the gearbox oil been changed? Check the driveshaft to the front diff and check the oil level in both diff and the outboard hubs.
It's really not a throttle dampner it is for the hst to help prevent the sudden stop if your foot slips off the peddle. My advice is either leave it or remove and throw away and get a coast valve
The tank under the seat is the common reservoir for the hst the power steering and the bed lift. The gear case is it's own reservoir and has no pump or filters. The gear case is splash lube.
Take out relay and jump 12 volt from the relay plug to the wire leading to fan motor... if fan runs then check the 12 volt supply to the relay plug? If that passes swap relay with another and test again. If that fails you have a bad ground or the wire from sender to relay plug is bad
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If your charge pressure is low it will still move but sluggish. I have heard of some losing prime or the charge pump not pumping then no movement. To test the charge pump you need to make up a Guage to test.