a Casimir invariant: $\gamma ^{(i)} = 0$ where $i$ varies from $1$ to $n.$ A system of equations, like this one, given by setting a collection of one-forms to zero, is called a Pfaffian system. The level sets of the functions on which the restrictions of these one-forms vanish are called the integral manifolds of the system.

Finding the integral manifolds is conceptually (but not always computationally) simple. If we were able to linearly combine the $\gamma
^{(i)}$ into $n$ independent exact differentials, we would obtain a system of equations equivalent to $\gamma ^{(i)} = 0$. This system would have the form ${\rm d}F^{(j)} = 0$ where $j$ varies from $1$ to $n.$ The solution to this system of equations is obviously given by $F^{(j)} = {\rm constant}.$ In other words, the integral manifolds are just the level sets of the $%
F^{(j)}.$ By the chain rule, we can then write the general solution of $%
\gamma ^{(i)} = 0$ in terms of an arbitrary function $k = k(F^{(1)},\cdots
,F^{(n)}).$

It is not always possible to transform a Pfaffian system in this way. The conditions for when it is possible are given by the Frobenius theorem. In [@frob] it was 