On some commutative subalgebras of the universal enveloping algebra of the Lie algebra $\mathfrak{gl}(n,\mathbb C)$

For the Lie algebra $\mathfrak g=\mathfrak{gl}(n,\mathbb C)$ it is proved that the maximal commutative subalgebras of the Poisson algebra $P(\mathfrak g)$ obtained by the method of shifting the invariants can be lifted to the enveloping algebra. Moreover, this lifting can be carried out by means of the symmetrization map.