Inspired by the Lie algebra discussed in this answer, we consider the following Lie agebra associated to a given foliation:

Let $\mathcal{F}$ be a nontrivial foliation of a manifold $M$ tangent to integrable subbundle $D$ of $TM$. We define the following Lie algebra of vector fields on $M$:

$$A_{\mathcal{F}}=\{Y\in \chi^{\infty}(M)\mid [X,Y]\;\; \text{is tangent to $\mathcal{F}$} \text{ for all } X\in \Gamma (D)\} $$

In fact $A_{\mathcal{F}}$ is the idealizer of the Lie algebra $L_{\mathcal{F}}$ of vector fields on $M$ which are tangent to the foliation.

First Question: Is there a foliation $\mathcal{F}$ for which $A_{\mathcal{F}}=L_{\mathcal{F}}$?

The second question:Is it true to say that the dimension of $\{Y(p)\mid Y \in A_{\mathcal{F}}\}\subseteq T_pM$ is independent of Choosing $p\in M$?

If the answer of the second question is yes, then $A_{\mathcal{F}}$ defines an integrable distribution $D'$ containing the initial distribution $D$. It generates a foliation $\mathcal{F}'$ which would be defined as saturation of $\mathcal{F}$.

Is there an example of this situation such that $dim(\mathcal{F}) <dim({\mathcal{F}')<dim(M)}$?

**Remark:** As we see in the linked question, when we have a $1$-dimensional foliation $\mathcal{F}$ tangent to a non vanishing vector field $X$ on a surface $M$ with volume form $\omega$ , then the Lie algebra $A_{\mathcal{F}}$ is equal to $$\{Y\mid [X,Y]=fX,\;\text{for some }f\in C^{\infty}(M)\}=\{Y\in X^{\infty}(M)\mid X.\omega(X,Y)=Div X\omega(X,Y)\}$$.

**Added:** According to the comment of Bertram Arnold we add the following question:

Is it true to say that there is an open dense subset $U\subset M$ with the following two properties:

1) For every $x\in U,\; \{V(x)|V\in A_{\mathcal{F}}\}=T_x M \}$.

2)$U$ and $M\setminus U$ are $\mathcal{F}$- saturated.

Then it seems that $M\setminus U$ is a characyteristic set in the sence that it is invariant under every leaf preserving diffeomorphism.