Integrable deformations of Rikitake systems, Lie bialgebras and bi-Hamiltonian structures

The paper proves that for the Poisson pencil [X,Y]_λ = (1−λ) 2β W + 2λ Z, [Y,Z]_λ = X, [Z,X]_λ = (−1 + 2λ) Y, [W,·]_λ = 0 (β ≠ 0), any λ‑independent 1‑cocycle δ ultimately fails the co‑Jacobi identity unless δ ≡ 0. They set up a general skew pre‑cocommutator δ with 24 coefficients, impose the 1‑cocycle equations for the entire pencil, and reduce to a restricted form (their Eqs. (4.16)–(4.21)); the remaining λ‑independent candidate is then killed by co‑Jacobi, yielding only the trivial cocommutator (Theorem 1) . The candidate solution asserts a 2‑parameter family of common 1‑cocycles and claims co‑Jacobi enforces p^2 = q^2, so that nontrivial δ exist. Re‑deriving and checking the candidate family shows: (i) it does satisfy the 1‑cocycle identity for all λ; however, (ii) the co‑Jacobi identity forces p = q = 0 (over ℝ, β ≠ 0), contradicting the candidate’s claim. This matches the paper’s conclusion that no nontrivial bi‑Hamiltonian Poisson–Lie integrable deformation exists for β ≠ 0 . A minor technical note: in obtaining (4.20) the paper sets b6 = c4 = 0 to remove a λ = 0 pole; a cancellation condition b6 = 2β c4 would also remove this singularity and yields a slightly larger 2‑parameter space of λ‑independent 1‑cocycles before the co‑Jacobi test, but the co‑Jacobi identity still forces triviality. Hence the main result and conclusion of the paper remain correct.