A Pfaffian system is a collection, for 
, ..., 
, of first-order differential
equations
 |
(1)
|
where the 
are differential forms (Esteves and Kleiman
2003). A linear Pfaffian system for a vector of unknown functions 
can be written
 |
(2)
|
where 
is a matrix-valued differential
form. The matrices 
are called connection matrices. They determine the partial
derivatives of 
by 
.
A compatible linear Pfaffian system satisfies the flatness condition
 |
(3)
|
This follows from 
(Doran 2001). Here 
is the exterior derivative and 
is the wedge product. Restricting
a Pfaffian system to a one-dimensional path gives an ordinary
differential equation, which can be solved locally by the Frobenius
method (Banik and Bera 2026).
