  Let $\mathfrak g$ be a semisimple Lie algebra, $\vartheta\in {\sf
Aut}(\mathfrak g)$ a finite order automorphism, and $\mathfrak g_0$ the
subalgebra of fixed points of $\vartheta$. Recently, we noticed that using
$\vartheta$ one can construct a pencil of compatible Poisson brackets on
$\mathcal S(\mathfrak g)$, and thereby a `large' Poisson-commutative subalgebra
$\mathcal Z(\mathfrak g,\vartheta)$ of $\mathcal S(\mathfrak g)^{\mathfrak
g_0}$. In this article, we study invariant-theoretic properties of $(\mathfrak
g,\vartheta)$ that ensure good properties of $\mathcal Z(\mathfrak
g,\vartheta)$. Associated with $\vartheta$ one has a natural Lie algebra
contraction $\mathfrak g_{(0)}$ of $\mathfrak g$ and the notion of a good
generating system (=g.g.s.) in $\mathcal S(\mathfrak g)^\mathfrak g$. We prove
that in many cases the equality ${\mathsf{ind\,}}\mathfrak
g_{(0)}={\mathsf{ind\,}}\mathfrak g$ holds and $\mathcal S(\mathfrak
g)^\mathfrak g$ has a g.g.s. According to V.G. Kac's classification of finite
order automorphisms (1969), $\vartheta$ can be represented by a Kac diagram,
$\mathcal K(\vartheta)$, and our results often use this presentation. The most
surprising observation is that $\mathfrak g_{(0)}$ depends only on the set of
nodes in $\mathcal K(\vartheta)$ with nonzero labels, and that if $\vartheta$
is inner and a certain label is nonzero, then $\mathfrak g_{(0)}$ is isomorphic
to a parabolic contraction of $\mathfrak g$.
