rank(J) $= \; 2$ everywhere except at the origin. The null covector of J is immediately found to be $\gamma = z^1{\rm d}z^1 +
z^2\mbox{d}z^2 + z^3\mbox{d}z^3.$ And the Pfaffian system $\gamma = 0$ is immediately solved for the well-known Casimir invariant $C=
(z^1)^2+(z^2)^2+(z^3)^2 = \mbox{constant}.$

As a second example, we consider the brackets of arbitrary dimension $d$ introduced by Plank [@pla1] for Lotka-Volterra equations (see also [@byv3] for an application of such brackets to more general systems). In this case, the components of J are given by (not summed): $$\label{ND.1}
{\rm J}^{ij} = c_{(ij)} z^i z^j \quad \quad i,j = 1 \ldots d$$ where $c_{(ij)} \in I \! \! R$ and $c_{(ij)} = - c_{(ji)}$ for all $i,j$. It can be shown [@pla1] that the skew-symmetry of the $c_{ij}$ is enough to ensure that J in (\[ND.1\]) indeed defines a cosymplectic form. It is simple to prove that every null covector of the matrix J in (\[ND.1\]) is of the form:

$$\label{ND.2}
\gamma = a_i \frac{\mbox{d}z^i}{z^i} \, , \:\;\:\;\:\; a_i \in I \!\! R \, .$$

Every Pfaffian equation $\gamma =0$, with $\gamma$ given by expression (\[ND.2\]), can be integrated to give one Casimir invariant:

$$\label{ND.3}
C = a_i 