  We prove an analogue of Thurston's h-principle for $2$-dimensional foliations
on manifolds of dimension bigger or equal to $4$, in the presence of a
fiber-wise non-degenerate $2$-form. This helps us understand the flexibility of
rank $2$ regular Poisson structures on open manifolds with dimension bigger or
equal to $4$ and it also helps us understand the flexibility of Poisson
structures (not regular) on closed $4$-manifolds.
