Hi and welcome to FixYa,
Almost always, when an electronic device/equipment fails when powered with and by a generator, it is the power supply unit (PSU). The PSU circuitry expects a fairly constant/stable power source (120 or 220). The output of the generator tends to fluctuate especially under varying electrical load. Additionally, the AC frequency (60Hz) of the generator's output also tend to vary which most PSUs don't like. A noteworthy point to consider is that the generator is floating electrically, its output has no reference to earth/ground.
If you are considering a DIY (
do-it-yourself), foremost to inspect would be the PSU particularly the regulator transistor and determine if it outputs +5 and ~+135 VDC. Another to check would be the HOT (
horizontal output transistor). If when the TV was used powered by a generator, there was considerable vertical rolling, the vertical deflection IC would also be a suspect.
A friendly reminder, exercise caution when repairing TVs. It stores up EHV (
extremely high voltage). It would be to your advantage to reference a
service manual (JVC AV-36D303) or at the very least a
schematic diagram with voltage readings.
Good luck and Thank you for using FixYa.