This bracket is sometimes called the Lagrange bracket in the literature (although that terminology is unfortunately not universal, and sometimes refers to something different). It can be characterised as the map $\lbrace\cdot,\cdot\rbrace:C^\infty(M)\times C^\infty(M)\to C^\infty(M)$ that satisfies $[X_F,X_G] = X_{\{F,G\}}$ (where $X_F$ is the vector field on $M$ described in the linked question). The analogy one should consider is that contact form is to Lagrange bracket as symplectic form is to Poisson bracket induced by the symplectic form: the 'characteristic distribution' corresponding to the Lagrange bracket is the distribution spanned by vector fields $X_F$ for all $F\in C^\infty(M)$, and this is just the entire space $M$. So characteristic distributions of Lagrange brackets have one leaf.
The generalisation of the Lagrange bracket, analogous to a general Poisson bracket, is sometimes called the Jacobi bracket. From "On Jacobi Manifolds and Jacobi Bundles" by Marle:
Theorem [Kirillov]. The characteristic distribution of a Jacobi manifold $(M,\Lambda, E)$ is completely integrable in the sense of Stefan and Sussman, thus defines on $M$ a Stefan foliation (i.e., a foliation whose leaves may not be all of the same dimension). Each leaf of this foliation has a unique transitive Jacobi structure such that its canonical injection into $M$ is a Jacobi map.
Remark. The last theorem, together with remark 2.7.3, shows that each leaf of the characteristic foliation of a Jacobi manifold is
- a locally conformally symplectic manifold if its dimension is even,
- a manifold equipped with a contact 1-form if its dimension is odd.
I have posted this answer on math.stackexchange.com also.
