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Do you mean a Poisson bracket on the space of functions? Then there is no obstruction, as you can always take the Poisson bracket to constantly vanish.
Not sure what you need. The vanishing Poisson tensor has maximal kernel regarded as a map $\pi \colon \Gamma(T^\ast X) \longrightarrow \Gamma(T X)$. You could ask for a smaller kernel. If you ask for vanishing kernel, then you may want a symplectic manifold structure.
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