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” $\mathit{b}(\mathfrak{g})$ 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 $\mathfrak g_0\ltimes\mathfrak g_1^{\sf ab}$hasapolynomialringofsymmetricinvariants,then$\tilde{\mathcal Z}$isapolynomialmaximalPoisson-commutativesubalgebraof$S(\mathfrak g)^{\mathfrak g_0}$, and its free generators are explicitly described.