Maybe we can have a King/Queen just for show and a PM with some power and then the rest of the power lies with the branches below the PM.
I actually favor this one.
The King/Queen would have a role - knighting people, occasionally commenting on politics, visiting foreign countries, entertaining dignitaries, etc. Just not one that had any effect whatsoever.
Personally, I'd like the role - much more fun then being President, actually. Less responsibility, too, so I won't go and blow up the planet.
Example of what I mean.
Let's say I become King Daniel I and rule the nation powerlessly with a ham-fist for 5 months. As part of the role, I host a few parties, knight a few people, make Verin the Earl of Duluth, and a few other things. Then, I appoint Everett as my successor and abdicate, and she becomes Queen Everett I. She does her own thing, keep the royal rabbits in line, make Al Viscount of West Virginia, throw a few *hughughug* parties, and then after four and a half months tires of it, names Verin her successor and abdicates. He becomes King Verin I, he rules for six months, appoints PBrunsel who becomes King Paul I, etc...