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It is shown that the $gl(3)$ polynomial integrable system introduced bySokolov-Turbiner is equivalent to the $gl(3)$ quantum Euler-Arnold top in aconstant magnetic field. Their Hamiltonian and their 3rd order Integral can berewritten in terms of $gl(3)$ algebra generators. All $gl(3)$ generators can berepresented by the non-linear elements of the universal enveloping algebra ofthe 5-dimensional Heisenberg algebra $h_5(\hat{p}_{1,2},\hat{q}_{1,2}, I)$.Four different representations of the $h_5$ Heisenberg algebra are used: (I) bydifferential operators in two real (complex) variables, (II) byfinite-difference operators on uniform or exponential lattices. We discoveredthe existence of two 2-parametric bilinear and trilinear elements (denoted $H$and $I$, respectively) of the universal enveloping algebra $U(gl(3))$ such thattheir Lie bracket (commutator) can be written as a linear superposition of {\itnine} so-called {\it artifacts} - the special bilinear elements of $U(gl(3))$,which vanish once the representation of the $gl(3)$-algebra generators iswritten in terms of the $h_5(\hat{p}_{1,2},\hat{q}_{1,2}, I)$-algebragenerators. In this representation all nine artifacts vanish, two of theabove-mentioned elements of $U(gl(3))$ commute(!). If $(\hat{p},\hat{q})$ arerepresented by finite-difference/discrete operators on uniform or exponentiallattice, the Hamiltonian and the Integral of the 3-body elliptic Calogero modelbecome the isospectral, finite-difference operators on uniform-uniform orexponential-exponential lattices (or mixed) with polynomial coefficients. If$(\hat{p},\hat{q})$ are written in complex $(z, \bar{z})$ variables theHamiltonian corresponds to a complexification of the 3-body elliptic Calogeromodel on $C^2$.

$gl(3)$ polynomial integrable system: different faces of the 3-body/$A_2$ elliptic Calogero model