01 praire400 runs but is pushing oil out of crankcase to airbox?

IT is likely that the carb is leaking fuel and filling the crankcase with fuel. Causing excessive crankcase pressure forcing oil into airbox through the crankcase vent. Check your crankcase for fuel and/ or being overfilled. If fuel is present fix the carb first, then drain and refill crankcase. After all is fixed run the unit outside for a few minutes and let it burn off excess smoke.

air boxes usually have a hose coming out of them for just that... to drain the air box of oil or water that gets in it. it doesnt attach to anything, it's just to drain the box onto the ground instead of all over the bike.

It is quite normal to have a small amount of oil in the airbox, however exsess oil in the airbox will find its way into the engine and burn hence smoke, remove the engine breather pipe from the engine to airbox and clean then replace, exsess crankcase pressure is caused by pressure leaking past the piston or rings, during detanation, dismantling of the top end of the engine is required for inspection, poss new piston and rebore.

Your PCV (Positive Crankcase Valve) has failed. The valve is designed to allow gas fumes in the crankcase to be recycled back through the intake and be burned off in combustion. There should be a black hose running into the airbox. It should also have a filter in the airbox. If you follow the hose to the engine itself it will end at the PCV. Replace the valve and the small filter in the airbox and this should resolve your problem as well as boost your fuel mileage.

This hose should be connected to the air filter box.

It's for ventilation of the crankcase, much like the PCV (Positive Crankcase Ventilation) in cars. As the bike runs, the crankcase gets hot enough to vaporize some of the oil. And since there's always at least a little bit of combustion blowby in the cylinders, you get a bit of fuel/air thrown into the mix and an oil/fuel/air "atmosphere" in the crankcase, which isn't an ideal situation. So, when the hose is connected to the air filter box, it slightly vacuums off this atmosphere, letting the engine burn it and expel it through the exhaust.

If it's not connected to the airbox, the vapors can condense back to oil and just drip out.

If a lot of oil is coming out ("a lot" is hard to quantify), you could be getting excessive blowby because of a broken ring or cracked piston.

Hook that breather line back up to the airbox and unless the bike smokes excessively, I wouldn't worry about it.

If it IS hooked up, it or the airbox is cracked and needs fixing/replacing. If the breather line has somehow become plugged (mud-daubers are amazing at finding holes to makes homes out of), the pressure buildup could have split your vent line.

Could be many things, simplest would be if you overfilled the oil...

Yep, its somewhat normal. Your crankcase ventilation tube connects to the back of the airbox. There is a small foam to dampen the airflow and remove the droplets from the flow - but other than that the crankcase is connected to the airbox. Now the good question is that what amount is normal, i.e. under what conditions there may be excessive amounts of oil generating so much positive pressure to crankcase that the oil becomes clearly visible in airbox... one reason may be too high oil level to start with ? Other reasons may relate to some kind of piston blowby - like rings glued into the piston? Also it is sometimes possible that PAIR lets air to move back to the airbox - but have not seen that happening in a busa, just in a car. In that case rather than oil its black carbon residue which feels a bit oil'ish. You can test the crancase pressure by removing the hose from the back of the airbox and feel the pressure when engine is running. It should not be blowing air out - but some pressure, or more like pulses will be felt. (Its so long time when I have last done this test, so maybe someone could describe this better...) Anyway this reminded me about the fact that in cars there is often a PCV valve, positive crankcase ventilation valve. I am just wondering why we dont have that in busa ? The performance related desire to have negative crankcase pressure surely is not the reason...

Yep, its somewhat normal. Your crankcase ventilation tube connects to the back of the airbox. There is a small foam to dampen the airflow and remove the droplets from the flow - but other than that the crankcase is connected to the airbox. Now the good question is that what amount is normal, i.e. under what conditions there may be excessive amounts of oil generating so much positive pressure to crankcase that the oil becomes clearly visible in airbox... one reason may be too high oil level to start with ? Other reasons may relate to some kind of piston blowby - like rings glued into the piston? Also it is sometimes possible that PAIR lets air to move back to the airbox - but have not seen that happening in a busa, just in a car. In that case rather than oil its black carbon residue which feels a bit oil'ish. You can test the crancase pressure by removing the hose from the back of the airbox and feel the pressure when engine is running. It should not be blowing air out - but some pressure, or more like pulses will be felt. (Its so long time when I have last done this test, so maybe someone could describe this better...) Anyway this reminded me about the fact that in cars there is often a PCV valve, positive crankcase ventilation valve. I am just wondering why we dont have that in busa ? The performance related desire to have negative crankcase pressure surely is not the reason...

Deep sumps are to keep the pickup in the oil at all times even when the front end is way off the ground - I don't think it'll have any effect on sending oil out the crankcase breather. An aftermarket tail-mounted breather box will keep oil out of the airbox but it's expensive probably hard to find nowadays, and forces you to reroute a lot of wiring as well as relocate your computer box. Simpler to just do fewer and smaller wheelies,,,

Deep sumps are to keep the pickup in the oil at all times even when the front end is way off the ground - I don't think it'll have any effect on sending oil out the crankcase breather. An aftermarket tail-mounted breather box will keep oil out of the airbox but it's expensive probably hard to find nowadays, and forces you to reroute a lot of wiring as well as relocate your computer box. Simpler to just do fewer and smaller wheelies,,,