Mouseguard (a simpler adaptation of Burning Wheel) fits some of the bill. The system is really easy, conflict is abstract and narrative rather than turn-by-turn, and the game pretty much expects the players to state their goals and the GM to adapt and bend the session to those stated goals. It is a game though, where the GM is explicitly encouraged to go hard on the players and throw even the kitchen sink at them, challenge-wise - so I don't know where it stands on your "non adversarial" clause.

the games suggest a rather formulaic session structure that I find quite interesting, that goes like this:
1- GM introduces the current mission / task
2- each players declares a Goal tied to the session task
3- a GM turn, in which challenges are thrown at the players, and ends either with the resolution of the mission, or in a place where the characters can "slow down" and rest. The game assumes the GM will tailor the challenges to the Goals the players stated in step 2 above
4- a player's turn, in which the players get a number of "scenes" to either recover from consequences accrued during the GM turn, or to pursue personal goals.

plus, you are playing anthropomorphic, intelligent mice. It can be enough to sell the game to some audiences :D