Obstructions to maximal number of independent constants of motion in a given symplectic manifold

Question

Given a compact symplectic manifold $(X, \omega)$, are there any invariants (topological or easily computable geometric/analytic ones) which give an estimate of the maximal number of independent constants of motions in $(X, \omega)$?

More precisely, we would like to find a reasonable bound on a number $k \in \mathbb{N}$, which is the maximal number such that there are $k$ generically linearly independent functions $f_1, \ldots, f_k$ (i.e. $df_1 \wedge \cdots \wedge df_k \neq 0$ generically), which Poisson commute with respect to $\omega$.

Trivially, we have $1 \leq k \leq n$, where $\dim X = 2n$. Moreover, if $k = n$, then by Liouville-Arnold theorem the manifold $X$ admits a (Lagrangian) torus fibrations away from the singular points, which is a non-trivial topological obstruction (I suppose; is it by the way correct?).

Answer 1

Any symplectic 2n-dimensional manifold admits a systems of Poisson-commuting functions $f_1,..,f_n$ whose differentials are linearly independent on an open subset of full measure.

The result is proven in the book ''Symplectic geometry" of A.T. Fomenko. The initial russian version and its engish translation have different paging; in the english version you should start with the page 145.

The proof is as follows: one first constructs such functions $f_1,...,f_n$ on the standard symplectic space such that the support of each function $f_i$ is the same standard 1-ball
and then fills the manifolds by such balls, properly scaled.

Of course the singularities of such system are nasty and form a multidimentional Cantor-type set