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.

]]>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.

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