**Pfaffian Equations: **equation on of the kind , where .

We will discuss smooth distributions of plane fields on a dimensional manifold . Locally we can describe them as a quotient , where we mean that if and only if there exists an open neighborhood of and a diffeomorphisms on this neighborhood such that .

**Local structure of the generic plane field**

- Let be a topological space. We say that a subset is
*residual*if and only if , where the sets have the property that . - A property is
*generic*in if and only if it holds on a residual subset of . **Theorem (Thom)**Let be any vector bundle, for some . Assume that is a submanifold; then the set is residual.- We recall that if is a smooth map and is a submanifold, we say that if and only if we have .
**Corollary:**For the generic section , the set is a regular submanifold of codimension equal to

We apply this machinery to our case:

- if , then we have a contact structure, and all contact structures are locally equivalent by Darboux theorem.
- On the contrary, we may define the set , and see that it is a submanifold of codimension one. Pull it back on the manifold to obtain a surface, the Martinet surface, that is the set (regular for the generic section).
- In the condition , and the Martinet surface is the zero set of .
- Define a foliation tangent to , that is on the set and .
**Claim:**let be a smooth vector field on such that and if and only if . Then, whenever , in any local coordinate system we have . Indeed assume that are coordinates on . Then and . Moreover .

**Theorem (Zhitomirskii): **let be a generic never vanishing one-form. Then there exist a regular surface and two discrete sets such that

- if , then is locally equivalent to
- if then is locally equivalent to
- if then is locally equivalent to , for some
- if then is locally equivalent to , for some