Let $M$ be a compact smooth manifold. Since any vector field is complete we get a $1$-parameter subgroup for each vector field. Consider the following generalization:

Let $\{X_j\} \in Vect(M)$ be a finite "basis" of some integrable subbundle of $TM$ (meaning that their locally linearly independent and closed under lie bracket). We have $Span\{X_j\} \cong \mathfrak{g}$ for some finite dimensional lie algebra. The exponential map $\varphi : \mathfrak{g} \to G$ gives a group equipped with a natural action $\rho: G \to Diff(M)$. Denoting the flow of $X$ by $\varphi_t^X$ the action looks like:

$$\rho : g=e^{X_1}e^{X_2}\dots e^{X_n} \mapsto \varphi_1^{X_1}\circ e_1^{X_2}\circ\dots \circ\varphi_1^{X_n}(-)$$

Question 1: Is this map well defined?

In any case if $\mathfrak{g}$ is abelian the action is well define and we get a product of circles and lines inside $Diff(M)$. For every point $x \in M$ the action of the torus part will carve an embedded submanifold of $M$ and the action of the euclidean part will carve an immersed submanifold.

Question 2: Is there anything more substantial to say here? When will the action of the torus part yield an actual torus?

Question 3: Does this construction give a $G/\ker\rho$-fiber bundle?(does it help if $G$ is compact?).