R structures, Yang–Baxter equations, and related involution theorems

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There are several ways to construct functions in involution on a Lie bi-algebra, a Lie algebra equipped with a second Lie bracket. For the solvable systems associated to the Casimir functions a second Hamiltonian formulation can be constructed and a class of bi-Hamiltonian Korteweg–de Vries-like evolution equations with explicit space dependence is derived. Translating the Casimir functions with the flow of a special vector field yields another set of functions in involution. Lenard relations are found for the corresponding Hamiltonian systems. Finally, solutions of the classical Yang–Baxter equation lead to an analog of compatible Hamiltonian pairs. The invariants of the resulting hereditary operators are in involution.