(finite-dimensional) Poisson bracket defined on the space D. This will be referred to as the inner bracket. This bracket is an infinite-dimensional generalization of the Lie-Poisson bracket, first considered by Lie [@lie]. (The bracket is sometimes also known as the Kostant-Kirillov bracket [@koki].)

Furthermore, the brackets we consider can be rewritten in the form $$\label{INFCOSYM}
\{ F,G \} = -\int _{{\rm D}} {\frac{\delta F }{\delta f}} \left[ f, {\frac{%
\delta G }{\delta f}} \right] {\rm d}^n z \, .$$ This is not true for all brackets of the form (\[INFBRAC\]). Whether it holds or not depends on the nature of the inner bracket, as will be seen below.

Now, from equation (\[INFCOSYM\]), we can read off the cosymplectic operator as ${\rm J} = -[f, \cdot ] $. So the condition for finding the components $\gamma $ of the null covector $\Gamma \colon = \gamma \delta f$ of J is simply

$$\label{INFNULL}
[f, \gamma ] = 0.$$

To solve this completely for $\gamma $ we need to know the details of the inner bracket. These will be provided for two cases below.

If the inner bracket in (\[INFBRAC\]) is canonical, that is, takes the form (\[CANBRAC\]), an integration by parts (assuming 