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Extra info for Communications in Mathematical Physics - Volume 196

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Abelian Lie Algebras of Symmetries Let us consider a Tq -dense dynamical system with constant coefficients: I˙1 = 0, . . , I˙p = 0, ϕ˙ 1 = ω1 , . . , ϕ˙ q = ωq . 3). 1). 1) which does not depend on the choice of the toroidal coordinates I1 , . . , Ip , ϕ1 , . . , ϕq . 2) because vectors a1 , . . , ar ∈ Rq are linearly independent. 3) q − p ≤ v ≤ q − rm 42 O. I. 3). 1) if rm = p or p − 1. Then v = q − p = 2(q − k) and tori Tq are coisotropic for q = k + 1, . . , 2k and Lagrangian for q = k.

11) have the following exact solutions: 1 H = H0 − vr2 , 2 = − 4η 2 + ω2 r2 H0 , F = ±ωH0 , H0 = sin(ω(z − vt)) sin(ηr2 ), where v, η, ω are arbitrary parameters. 13) t˙ = 1. The fluid velocity V is smooth and bounded everywhere in R4 . 13) is steady and has infinitely many invariant compact domains which are separated by invariant surfaces H0 (t, z, r) = 0 : z − vt = nπ/ω and r2 = mπ/η. Here n and m are arbitrary integers and m/η > 0. In each domain, trajectories are either quasi-periodic on tori T2 defined by the equation H0 (t, z, r) = const = 0, 0 ≤ ϕ ≤ 2π, or are circles z − vt = (n + 1/2)π/ω, r2 = (m + 1/2)π/η, ϕ˙ = ±(−1)n+m ω/r2 .

32 O. I. Bogoyavlenskij Definition 4. 1) is called Tq -dense in the toroidal domain O = Ba × Tq ⊂ M n if the set X is everywhere dense in the ball Ba . For example, a Liouville-integrable Hamiltonian system on a symplectic manifold M 2k is Tk -dense if it is non-degenerate in the Poincar´e sense or in the Poincar´e iso-energetic sense [9, 31]. Proposition 3. 1) is Tq -dense in the toroidal domain O = Ba × Tq ⊂ M n if and only if the functions ωα (I) are rationally independent in any ball B0 ⊂ Ba .