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Discussed in the present study are details of the behavior of low-frequency "acoustic" ($ac$) modes in the spectrum of edge magnetoplasma oscillations in axially symmetric degenerate 2D electron systems where electron density distribution $n(\vec r)$ behaves as $n(\vec r \to R)\to 0$ when $r$ approaches the external radius $R$ of the domain occupied by electrons. It is shown that finding the dependence of the spectrum of $ac$-modes on radial ($l$) and azimuthal ($m$) indices when both $l$ and $m$ are small requires axially symmetric solution of the relevant problem. The desires is achieved in the so-called elliptic approximation for electron density distribution $n(\vec r)$. The obtained results are employed to interpret available data on the excitation of $ac$-modes in degenerate electron disks with smooth electron density profile placed in the magnetic field $H$ normal to the disk plane. The performed analysis confirms reported detection of the soft $ac$-mode in the range of $H \gg H_{max}$ where $H_{max}$ is the field at which the maximum of the $\omega_{lm}^{s}(H)$ curve is observed (here and below $\omega_{lm}^{s}(H)$ stands for the frequency of the soft $ac$-mode). A strong interaction of the $ac$-modes with $(integer)$-channels inevitably arising near the boundaries of 2D electron systems with smooth electron density profile as the magnetic field is varied in the Quantum Hall Effect (QHE) regime is emphasized. Well-formed $integer$-stripes can suppress some acoustic modes which is actually observed in experiments.