Let $(M,\omega)$ be a finite-dimensional symplectic manifold. The algebra $C^\infty(M)$ of smooth functions is a Poisson algebra. Derivations $D : C^\infty(M) \to C^\infty(M)$ of the Poisson structure are naturally identified with vector fields which preserve the Poisson bivector, which in the symplectic case is equivalent to being a symplectic vector field.
In this question, I am however interested not in Poisson derivations, but only in Lie derivations; that is, real linear maps $D : C^\infty(M) \to C^\infty(M)$ which obey $$ D\{f,g\} = \{Df,g\} + \{f,Dg\}$$ for all $f,g \in C^\infty(M)$. In particular, since $D$ is not necessarily a derivation of the commutative algebra structure, so that $D(fg) - (Df)g - fDg$ need not be zero, $D$ is not necessarily a vector field.
Have such Lie derivations been studied? And, if so, where?
