Inspired  by the  Lie algebra discussed   [in this   answer](https://mathoverflow.net/questions/193650/a-vector-space-associated-with-a-vector-field-on-a-symplectic-manifold/303162#303162),  we  consider the  following  Lie  agebra  associated to  a  given foliation:

Let  $\mathcal{F}$  be  a  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](https://mathoverflow.net/questions/302578/a-non-integrable-distribution-arising-from-a-lie-algebra-of-vector-fields/302579#302579) $D'$ 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 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 $x\in U,\; dim \{V(x)|V\in A_{\mathcal{F}}=dim M$.

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.