How large is the subgroup of diffeomorphisms which preserves a vector field?

Question

Let $M$ be a smooth manifold. Let $X \in \Gamma(TM)$ be a vector field on $M$, which vanishes at a finite number of points. (Every smooth manifold admits such a vector field).

Consider the subgroup $\{\phi \in \operatorname{Diff}(M)| \phi_*X=X \}$?

Is it always a finite dimensional Lie group? Or can it be infinite-dimensional?

Answer 1

It may well be infinite dimensional. Take on the torus the vector field $X =\frac{\partial}{\partial x}$. For any $g(y)$ smooth and periodic function, the vector field $g(y) \frac{\partial}{\partial x}$ satisfies $[Y,X]=0$, so the difeomorphism induced by $Y$ ( $1$-parameter group in fact) invariates $X$.