Once the engine warms up. and i stop, it won't restart!

If the engine is warm it should be pushed in, or turned down.
If the engine is cold it should be pulled out, or turned up.

Do any lights come on when you turn the key? Do you need to pull in the clutch leaver to start? Make sure engine kill switch is in run position, Make sure you have petrol if more than 2 months old change it. Check main and ignition fuses are OK in fact check all fuses while your at it. Make sure your battery is OK your charger may say full but that does not mean the battery has enough oomph to start the bike.
How long ago since it last ran OK does anything come to mind when you stopped it then. Times of no run = times of problems.

If your bike sits outside it can be moisture build up inside the engine. White oil indicates water in the oil. I would recommend changing the oil prior to riding the bike. Make sure that you check the coolant level to verify that it is not low. Warm the engine up and then drain the oil. The will help flush the moisture out of the engine. Once you have new oil in the engine go for a ride and check the oil again. If you still have moisture in the engine it may have additional problems such as water pump seal or even a head gasket issue.

Thanks and I hope that this helps.
Steve

when ordering parts for my zx6r g1 it has been pointed out to me that the g1-g2 & j1 part no's are the same and that they are the same bike, Search the Cornwall Kawasaki Centre parts and online manual/ parts fiche'

depending on your modle of course is it?

ZX636C6F Ninja ZX-6R (2006)
or
ZX636D6F Ninja ZX-6R (2006)

be carefull toc hose properly./ take your time and you will be happy with your order.

Go To The Site

this is what i found for you.

Clean your tank and rinse the petrol pump, check if petrol pump is running when switching ignition on. You answered your own question. The fuel is not reaching the carb from the tank

Without the manual, I would ensure they are set by holding carby vertical so that the float is hanging downward. In this position you shouldnt be able to blow into carb through petrol line and usually the float jointing will lie parallel to carb body

Use 10W40 semi synthetic oil, filler is on the right hand side engine case. Check level in the sight glass below the filler cap with bike stood upright.

its on top of the trans over the sprocket