Given a smooth manifold $M$, one can consider the Lie algebra $\mathcal{X}(M)$ of vector fields equipped with the standard Lie bracket. This is a standard machinery of differential geometry. Gelfand and Fuchs defined the Lie algebra of formal vector fields at $0 \in \mathbb{R}^n$ as linear combinations $$\sum_{j=1}^np_j(x_1,...,x_n)e_j$$ where $e_j$ is a standard basis of $\mathbb{R}^n$ and $p_j$ are formal power series in variables $x_1,...,x_n$. This definition is at the given point $0$ in $\mathbb{R}^n$, so formal vector fields are not ,,globally'' defined.

Is it possible to define formal vector fields globally on a given manifold $M$?

Also I would like to know

What is the significance of formal vector fields? For example, do they naturally arise as a Lie algebra of some natural (infinite dimensional) group?