you should do a compression test before you do anything else write down the pressure you get from each cylinder and then remove the head if the compression is low, like below 100~125 on both cylinders it should be around 150 on each side....
your valves could be too tight but that is not too likely it's probably got bad rings in it and it will need a new set of rings .
sounds like you need to do a complete tuneup on it first to determine what the problem is but at a guess it's rings !
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what do you have and how is it setup ?
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burning oil is a real good sign of Low compression ( oil getting by the rings and being burnt with the fuel) or valves too tight and bad valve guide seals, but that is fairly unlikely. but it could make it Look like it's got bad rings in it.....
smoking out one side can be a bunch of different problems like a stuck float in the carb, a busted ring on that side or other things....
if it's a points type machine it is possible the timing /point gap has slipped making that cylinder fire out of proper timing making it light on that side .
seriously, do a complete tuneup on it, timing ,valve adjust and change plugs and then do a compression test
Choppers have the problem of lack of maintenance guys just get on them and ride and never put time in on doing the maintenance
after a while it adds up and you get a failure....
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I'm betting it needs Rings but all I know so far is what you described ..... a soft exhaust on one side, and black plugs and smoke
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the missing can be from oil-foweled spark plugs so that really doesn't tell us much !, but checking timing and carb syncronization is alwayse a good thing to do.
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so what are you working on exactly ?
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Bob.........