Poisson-commutative subalgebras of $S(\mathfrak g)$ associated with involutions.
The symmetric algebra $S(\mathfrak g)$ of a reductive Lie algebra $\mathfrak g$ is equipped with the standard Poisson structure, i.e., the Lie-Poisson bracket. Poisson-commutative subalgebras of $S(\mathfrak g)$ attract a great deal of attention, because of their relationship to integrable systems and, more recently, to geometric representation theory. The transcendence degree of a Poisson-commutative subalgebra ${\mathcal C}\subset S(\mathfrak g)$ is bounded by the "magic number" $\boldsymbol{b}(\mathfrak g)$ of $\mathfrak g$. The "argument shift method" of Mishchenko-Fomenko was basically the only known source of $\mathcal C$ with ${\rm trdeg\,}{\mathcal C}=\boldsymbol{b}(\mathfrak g)$. We introduce an essentially different construction related to symmetric decompositions $\mathfrak g=\mathfrak g_0\oplus\mathfrak g_1$. Poisson-commutative subalgebras $\mathcal Z,\tilde{\mathcal Z}\subset S(\mathfrak g)^{\mathfrak g_0}$ of the maximal possible transcendence degree are presented. If the $\mathbb Z_2$-contraction $\mathfrak g_0\ltimes\mathfrak g_1^{\sf ab}$ has a polynomial ring of symmetric invariants, then $\tilde{\mathcal Z}$ is a polynomial maximal Poisson-commutative subalgebra of $S(\mathfrak g)^{\mathfrak g_0}$, and its free generators are explicitly described.