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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 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.

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.

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 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 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.

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 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.

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 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 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.

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 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 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.