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Examples of nearly integrable systems on $\mathbb{A}^3$ with asymptotically dense projected orbits

Given an integer $κ\geq2$, we introduce a class of nearly integrable systems on $\mathbb{A}^3$, of the form $$ H_n(θ,r)=\frac12 \Vert r\Vert ^2+\tfrac{1}{n} U(θ_2,θ_3)+f_n(θ,r) $$ where $U\in C^κ(\mathbb{T}^2)$ is a generic potential function and $f_n$ a $C^{κ-1}$ additional perturbation such that $\Vert f_n\Vert_{C^{κ-1}(\mathbb{A}^3)}\leq \tfrac{1}{n}$, so that $H_n$ is a perturbation of the completely integrable system $h(r)=\frac12\Vert r\Vert ^2$. Let $Π:\mathbb{A}^3\to\mathbb{R}^3$ be the canonical projection. We prove that for each $δ>0$, there exists $n_0$ such that for $n\geq n_0$, the system $H_n$ admits an orbit $Γ_n$ at energy $\frac12$ whose projection $Π(Γ_n)$ is $δ$-dense in $Π(H_n^{-1}(\tfrac{1}{2}))$, in the sense that the $δ$-neighborhood of $Π(Γ_n)$ in $\mathbb{R}^3$ covers $Π(H_n^{-1}(\frac{1}{2}))$.