Dual \(R\)-matrix integrability (Q1002613)

In this paper the authors propose another way, the so-called dual \(R\)-matrix integrability, to construct algebras of mutually commuting first integrals of some Euler-Arnold-type equations using the notion of the classical \(R\)-operator. Considered 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 are described in detail. The authors answered the natural question when the constructed commutative algebras also commute with respect to the \(R\)-bracket and briefly discussed the Euler-Arnold-type integrable equations for which the constructed commutative functions constitute the algebra of first integrals.