Kawasaki did have a Factory MotoGP team until the end of 2008 if I recall correctly - when the banks f*cked the world's economies KHI (Kawasaki Heavy Industries) had to make cutbacks and a MotoGP effort was something that they decided they didn't need to do under the circumstances. Unfortunately for them, they were contracted to provide a bike on the grid for the next season and breaching the obligation to Dorna would have cost them more than putting something out there so they managed to come to a bit of a compromise agreement by giving the new bike for 2009 to a team run under the Hayate banner - the bike ran in black (rather than green) and there was only enough money and parts to run a single rider, so of the two riders lined up for the factory contract that year, they released John Hopkins and retained Marco Melandri to ride the Hayate bike.
It didn't start off too badly, and I think Melandri finished 2nd at Le Mans in the race that was disrupted by the weather (when Rossi fell off after taking his dry bike back out, returned for repairs, got nicked for speeding in the pitlane and had to do a drive through penalty and ended up finishing last). Sadly for Melandri, the bike wasn't developed throughout the season as it remained just as it was when they got it (unlike all the others on the grid that received engine, chassis and electronic upgrades throughout the season) and it became less and less competitve and as a result he ended up trying harder and harder to compensate for it, resulting in the inevitable crashes that ate into their diminishing stocks of spares...

That's a very potted history of why there isn't a Factory Kawasaki in MotoGP at the moment, hope it helps.