Consider a symplectic manifold $(M, \omega)$. Let us define a concept of a complete set of observables:
A set of functions $f_i : M \to \mathbb R$ form a complete set of observables if any function which Poisson commutes (has vanishing Poisson brackets) with all of them is a constant.
That is the set above can feel the behaviour of a given function in all the directions.
I wonder whether specifying a values of all of $f_i$ will define a unique point on a manifold? That is do such functions form coordinates? If this is false, I'd like to see a simple counterexample. Also if generally the statement is false, is there probably a large class of situations when it is true?
This question is closely related to my question at Physics.SE.