  Let $\mathfrak q$ be a finite-dimensional Lie algebra, $\vartheta\in
Aut(\mathfrak q)$ a finite order automorphism, and $\mathfrak q_0$ the
subalgebra of fixed points of $\vartheta$. Using $\vartheta$ one can construct
a pencil $\mathcal P$ of compatible Poisson brackets on $S(\mathfrak q)$, and
thereby a `large' Poisson-commutative subalgebra $Z(\mathfrak q,\vartheta)$
consisting of $\mathfrak q_0$-invariants in $S(\mathfrak q)$. We study one
particular bracket $\{\,\,,\,\}_{\infty}\in\mathcal P$ and the related Poisson
centre ${\mathcal Z}_\infty$. It is shown that ${\mathcal Z}_\infty$ is a
polynomial ring, if $\mathfrak q$ is reductive.
