It is a Pfaffian system generated by a single 1-form, $\{\omega\}$, in a manifold $M$. Usually it is denoted by
$$ \omega \equiv 0 $$Example
$$ x y dx+zdy+z^2 y dz=0 $$$\blacksquare$
Pfaff's problem is to determine its integral manifolds of maximal dimension (@bryant2013exterior page 15).
If the Pfaffian system is completely integrable then the 1-form is called Frobenius integrable. It is a weaker condition that being closed. That is, a 1-form is Frobenius integrable if there exists $\mu$ such that $\mu \omega$ is closed. The function $\mu$ is an integrating factor.
To solve a Pfaffian equation it is usually used a property of the wedge product.
The integer $r$ defined by
$$ (d\omega)^r \wedge \omega\neq0, \quad (d\omega)^{r+1} \wedge \omega = 0 $$ is called the rank of the Pfaffian equation, or Pfaff rank of $\omega$. It depends on the point $x\in M$. In the case of constant rank it can be applied the Pfaff-Darboux theorem.________________________________________
________________________________________
________________________________________
Author of the notes: Antonio J. Pan-Collantes
INDEX: