In this section, we consider a distribution **D** = 〈*X*
_{1},⋯, *X*
_{
m
}〉 = 〈*ω*
^{1},⋯,*ω*
^{
n
}〉 on manifold *M*
^{
n + m
}.

### Definition 4.1

A diffeomorphism *F* : *M* → *M* is called a *symmetry of*
**D** if *F*
_{∗}
**D**
_{
x
}= **D**
_{
F(x)}for all *x* ∈ *M*.

Therefore,we have the following theorem.

### Theorem 4.2

The following conditions are equivalent:

- (1)

- (2)
*F*
^{∗}
*ω*
^{
i
}s determine the same distribution **D**; that is **D** = 〈*F*
^{∗}
*ω*
^{1},⋯,*F*
^{∗}
*ω*
^{
n
}〉;

- (3)
*F*
^{∗}
*ω*
^{
i
}∧⋯∧*ω*
^{
n
}= 0 for *i* = 1,⋯,*n*;

- (4)
, where *a*
_{
ij
}∈ *C*
^{
∞
}(*M*);

- (5)
(*F*
_{∗}
*X*
_{
i
}|_{
x
}) ∈ **D**
_{
F(x)}for all *x* ∈ *M* and *i* = 1,⋯,*n*; and

- (6)
, where *b*
_{
ij
}∈ *C*
^{
∞
}(*M*).

### Theorem 4.3

If *F* be a symmetry of **D** and *N* be an integral manifold, then *F*(*N*) is an integral manifold.

### Proof

*F* is a diffeomorphism; therefore, *F*(*N*) is a sub-manifold of *M*. From other hand, if *x* ∈ *N*, then *ω*
^{
i
}|_{
F(x)}= (*F*
^{∗}
*ω*
^{
i
})|_{
x
}= 0 for all *i* = 1,⋯,*n*; therefore, *F*(*N*) = {*F*(*x*) | *x* ∈ *N*} is an integral manifold. □

### Theorem 4.4

Let **N** be the set of all maximal integral manifolds and *F* : *M* → *M* be a symmetry, then *F*(**N**) = **N**.

### Proof

If *x* ∈ **N**, then *ω*
^{
i
}|_{
F(x)}= (*F*
^{∗}
*ω*
^{
i
})|_{
x
}= 0 for all *i* = 1,⋯,*n*; therefore, *F*(*x*) ∈ **N** and *F*(**N**) ⊂ **N**. □

Now, if *y* ∈ **N**, then there exists *x* ∈ *M* such that *F*(*x*) = *y*, since *F* is a diffeomorphism. Therefore, (*F*
^{∗}
*ω*
^{
i
})|_{
x
}= *ω*
^{
i
}|_{
F(x)}= *ω*
^{
i
}|_{
y
}= 0 for all *i* = 1,⋯,*n*; thus, *x* ∈ **N** and **N** ⊆ *F*(**N**).

### Definition 4.5

A vector field *X* on *M* is called an *infinitesimal symmetry of distribution*
**D**, or briefly a *symmetry* of **D**, if the flow
of *X* be a symmetry of **D** for all *t*.

### Theorem 4.6

A vector field

*X* ∈

**X**(

*M*) is a symmetry if and only if

### Proof

Let

*X* be a symmetry. If Ω =

*ω*
^{1}∧⋯∧

*ω*
^{
n
}, then {(Fl

^{
X
})

^{∗}
*ω*
^{
i
}}∧ Ω = 0, by condition (3) in Theorem 4.2. Moreover, by the definition

, one gets

□

Therefore **L**
_{
X
}
*ω*
^{
i
}|_{
D
}= 0.

In converse, let

**L**
_{
X
}
*ω*
^{
i
}|

_{
D
}= 0 or

for

*i* = 1,⋯,

*n* and

*b*
_{
ij
}∈

*C*
^{
∞
}(

*M*). Now, if

, then

where

and

Therefore, *γ* = (*γ*
_{1}⋯,*γ*
_{
n
}) is a solution of the linear homogeneous system of ODEs (2) with the initial conditions (1), and *γ* must be identically zero.

### Theorem 4.7

*X* is symmetry if and only if for all *Y* ∈ **D**, then [*X*,*Y*] ∈ **D**.

### Proof

By the above theorem, *X* is a symmetry if and only if for all *ω* ∈ A*nn*
**D**, then **L**
_{
X
}
*ω* ∈A*nn*
**D**.

The Theorem comes from the Theorem 2.6 (b): **L**
_{
X
}
*ω* = −*ω*∘**L**
_{
X
}on **D**. In other words, (**L**
_{
X
}
*ω*)*Y* = −*ω*[*X*,*Y*] for all *Y* ∈ **D**. □

Denote by Sym_{
D
} the set of all symmetries of a distribution **D**.

### Example 4.8

**(Continuation of Example 3.4)** Let

*k* = 2. A vector field

is an infinitesimal symmetry of

**D** if and only if

*L*
_{
Y
}
*ω*
^{
i
}≡ 0

*mod*
**D**, for

*i* = 1,2. These give two equations:

### Example 4.9

**(Continuation of Example 3.5)** We consider the point infinitesimal transformation:

Then,

*Z* is an infinitesimal symmetry of

**D** if and only if

*L*
_{
Z
}
*ω*
^{
i
}≡ 0

*mod*
**D**, for

*i* = 1,2,3. These give ten equations:

Complicated computations using Maple show that

and

*X* =

*X*(

*x*,

*u*−

*qy*),

*Y* =

*Y*(

*y*,

*u*−

*px*), and

*U*(

*x*,

*y*,

*u*) must satisfy in PDE: