Dual $R$-matrix integrability

Using the $R$-operator on a Lie algebra $\mathfrak{g}$ satisfying the modified classical Yang–Baxter equation, we define two sets of functions that mutually commute with respect to the initial Lie–Poisson bracket on $\mathfrak{g}^*$. We consider examples of the Lie algebras $\mathfrak{g}$ with the Kostant–Adler–Symes and triangular decompositions, their $R$-operators, and the corresponding two sets of mutually commuting functions in detail. We answer the question for which $R$-operators the constructed sets of functions also commute with respect to the $R$-bracket. We briefly discuss the Euler–Arnold-type integrable equations for which the constructed commutative functions constitute the algebra of first integrals.