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Let $(M,\{ ,\} )$ be a Poisson manifold with a vector field $X$. I define a morphism $\nabla_X$ of the Poisson brackets:

$$\nabla_X: {\cal C}^{\infty}(M) \rightarrow {\cal C}^{\infty}(M)$$

$$\nabla_X (f+g)=\nabla_X (f)+\nabla_X (g)$$

$$\nabla_X (\{ f,g \} )= \{ X(f), \nabla_X (g) \}+ \{ \nabla_X (f) ,X(g)\}$$

What is the space of these morphisms of Poisson brackets?

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