Examples of nearly integrable systems on A³ with asymptotically dense projected orbits

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Given an integer κ≥2, we introduce a class of nearly integrable systems on A³, of the form Hₙ(θ,r)=12 r ²+1n U(θ₂,θ₃)+fₙ(θ,r) where U∈ Cκ(T²) is a generic potential function and fₙ a Cκ⁻¹ additional perturbation such that fₙκ⁻¹₍ₐ₃₎≤ 1n, so that Hₙ is a perturbation of the completely integrable system h(r)=12 r ². Let Π:A³³ be the canonical projection. We prove that for each δ>0, there exists n₀ such that for n≥ n₀, the system Hₙ admits an orbit Γₙ at energy 12 whose projection Π(Γₙ) is δ-dense in Π(Hₙ⁻¹(12)), in the sense that the δ-neighborhood of Π(Γₙ) in R³ covers Π(Hₙ⁻¹(1/2)).