be guessed, this becomes difficult for large systems. The reader is referred to [@byv2] for additional examples.

Casimir invariants of infinite-dimensional brackets {#S-INFINITE}
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For infinite-dimensional examples, we first consider Poisson brackets that occur in the Hamiltonian formulation of the Vlasov-Poisson system and the 2D Euler equation [@olv3; @maip]. Then, we consider a modification of our first example that might arise in other types of kinetic theories, such as that which describes a spin gas.

All of the examples for this section arise in systems that can be described by a single field variable $f.$ This field is taken to be a function on an $n
$-dimensional space D, with coordinates $z^i$, and of time. In these examples, the space D is endowed with a Poisson bracket.

The brackets in these examples have the following general form: $$\label{INFBRAC}
\{ F,G \} [f] = \int _{{\rm D}} f \left[ {\frac{\delta F }{\delta f}}, {%
\frac{\delta G }{\delta f}} \right] {\rm d}^nz \, .$$ In this expression, The variables $F, G$ represent functionals of the field $%
f$, and $\delta F / \delta f$ denotes the functional derivative of $F$ with respect to $f$. The square brackets in the integrand denote the 