  It is shown that the $\mathfrak{gl}(3)$ polynomial integrable system,
introduced by Sokolov-Turbiner in [arXiv:1409.7439], is equivalent to the
$\mathfrak{gl}(3)$ quantum Euler-Arnold top in a constant magnetic field. Their
Hamiltonian as well as their third-order integral can be rewritten in terms of
$\mathfrak{gl}(3)$ algebra generators. In turn, all these $\mathfrak{gl}(3)$
generators can be represented by the non-linear elements of the universal
enveloping algebra of the 5-dimensional Heisenberg algebra
$\mathfrak{h}_5(\hat{p}_{1,2},\hat{q}_{1,2}, I)$, thus, the Hamiltonian and
integral are two elements of the universal enveloping algebra
$U_{\mathfrak{h}_5}$. In this paper, four different representations of the
$\mathfrak{h}_5$ Heisenberg algebra are used: (I) by differential operators in
two real (complex) variables, (II) by finite-difference operators on uniform or
exponential lattices. We discovered the existence of two 2-parametric bilinear
and trilinear elements (denoted $H$ and $I$, respectively) of the universal
enveloping algebra $U(\mathfrak{gl}(3))$ such that their Lie bracket
(commutator) can be written as a linear superposition of nine so-called
artifacts - the special bilinear elements of $U(\mathfrak{gl}(3))$, which
vanish once the representation of the $\mathfrak{gl}(3)$-algebra generators is
written in terms of the $\mathfrak{h}_5(\hat{p}_{1,2},\hat{q}_{1,2},I)$-algebra
generators. In this representation all nine artifacts vanish, two of the
above-mentioned elements of $U(\mathfrak{gl}(3))$ (called the Hamiltonian $H$
and the integral $I$) commute(!); in particular, they become the Hamiltonian
and the integral of the 3-body elliptic Calogero model, if $(\hat{p},\hat{q})$
are written in the standard coordinate-momentum representation.
