shown that the possibility of doing this with the set of null covectors of a finite-dimensional J follows from equation (\[JACOBI\]). Later, in [@kuro], an argument was given that extends this proof to continuum systems. However, there are some infinite-dimensional spaces in which the Frobenius theorem does not hold (see [@krmi]), so this should be seen as a formal argument.

Examples of applying the above process are given in the next two sections. In the finite-dimensional case, the greatest complication is finding integrating factors to make the resulting equations exact differential equations (see Section \[S-FINITE\]). The infinite-dimensional case is typically more complicated. Finding components of the null covectors requires solving partial differential equations, and it is not always obvious how to integrate the Pfaffian system.

Casimir invariants of finite-dimensional brackets {#S-FINITE}
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For our first example, we will find the Casimir invariant of a familiar Hamiltonian system: the free rigid body. The components of J are given by: $%
{\bf {\rm J}}^{ij} = \epsilon _k^{ij} z^k$ where $i,j = 1 \ldots 3,$ and $%
\epsilon$ denotes the Levi-Civita symbol, as usual. In fact, this bracket is the Lie-Poisson bracket associated to the Lie algebra so(3) (see Section \[S-INFINITE\]). Note that 