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Given an integer $\kappa\geq2$, we introduce a class of nearly integrablesystems on $\mathbb{A}^3$, of the form $$ H_n(\theta,r)=\frac12 \Vert r\Vert^2+\tfrac{1}{n} U(\theta_2,\theta_3)+f_n(\theta,r) $$ where $U\inC^\kappa(\mathbb{T}^2)$ is a generic potential function and $f_n$ a$C^{\kappa-1}$ additional perturbation such that $\Vertf_n\Vert_{C^{\kappa-1}(\mathbb{A}^3)}\leq \tfrac{1}{n}$, so that $H_n$ is aperturbation of the completely integrable system $h(r)=\frac12\Vert r\Vert ^2$.  Let $\Pi:\mathbb{A}^3\to\mathbb{R}^3$ be the canonical projection. We provethat for each $\delta>0$, there exists $n_0$ such that for $n\geq n_0$, thesystem $H_n$ admits an orbit $\Gamma_n$ at energy $\frac12$ whose projection$\Pi(\Gamma_n)$ is $\delta$-dense in $\Pi(H_n^{-1}(\tfrac{1}{2}))$, in thesense that the $\delta$-neighborhood of $\Pi(\Gamma_n)$ in $\mathbb{R}^3$covers $\Pi(H_n^{-1}(\frac{1}{2}))$.

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