2006 Yamaha Roadstar Silverado, Installed Roadburner Street Pro exhaust system and Dyno-jet stage one kit,now the rear cylinder keeps fouling out! Any suggestions??

I would check if it still has stock jets. you need stage 1 or 2 dyno jet kit. To get best out of that upgrades also look for any vacum leaka

I would recommend adjusting the idle screw on the carb. If it is not the idle adjustment, then check the choke as it may be sticking and that will cause the engine to idle at that rate.

nope you will get about 28 with the tuning you did
mine never got anything better than that and i dyno tuned it more than once

want a great source for roadstar information? try roadstarclinic.com. you might have a sticking exhaust valve in the rear cylinder! they are known for it when they have the pipes and jet kits installed! if that be the case, I have a fix for it that is a lot cheaper than pulling the heads!

The dealer is not interested.
The bike is set up very lean to meet emissions standards. Any of them will benefit from larger jets. It's probably running too lean.

No it will not. You probably don't NEED to rejet at all. If the manufacturer suggests it for your bike and pipe combination I would follow their advice. If they do not suggest it I would use a dyno to get the best performance from your new setup. Keep in mine your new exhaust only lowers the back pressure. You haven't changed the intake parameters at all as far as airflow. So any additional air going into the cylinders due to your new exhaust will only be because of the reduced volume in the cylinder while the intake valve is open. This is a rather small amount, so don't get all crazy and jump several jet sizes at once.

With the key off and the engine "running down", the idle circuit fuel is still going to into each cylinder then out the exhaust. A hot spot in the exhaust can ignite the unburned fuel creating an explosion, ( backfire ). Let the engine run down to an idle THEN turn the key off.

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Sounds like it may be over jetted, and running alittle rich. Can you see black smoke if you rev it up? Could be misadjusted floats after the kit was installed.