Let $A$ be a polynomial algebra over a field of characteristic $0$ in the variables $x_1,\dots,x_n$. Consider polynomials $f_1,\dots,f_m\in A$ and let $I$ be the ideal they generate in $A$. Moreover, assume that $\{\:,\:\}$ is a Poisson bracket on $A$, i.e., that it is a Lie bracket on $A$ and a derivation in each argument. We say that $I$ is a *Poisson ideal* in A if $\{I,A\}\subset I$. This is equivalent to saying that there exist polynomials $Z_{ij}^k\in A$ such that
\begin{align}
\{x_i,f_j\}=\sum_{k=1}^m Z_{ij}^k f_k,
\end{align}
where $j\in\{1,\dots m\}$ and $i\in\{1,\dots n\}$. In this case the bracket descends to $A/I$ and we say that $A/I$ is an *affine Poisson algebra*.

So far my attempts to construct examples with *nonzero* $Z_{ij}^k$ have failed. My question is: are there examples with nonzero $Z_{ij}^k$? In the case when the Poisson structure is constant or linear there might be conceptual reasons for the vanishing of the $Z_{ij}^k$. Does anybody know of results in this direction?