Geodesic
On some commutative subalgebras of the universal enveloping algebra of the Lie algebra $\mathfrak{gl}(n,\mathbb C)$
Sbornik. Mathematics, Tome 191 (2000) no. 9, pp. 1375-1382

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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. 
@article{SM_2000_191_9_a5,
     author = {A. A. Tarasov},
     title = {On some commutative subalgebras of the universal enveloping algebra of the {Lie} algebra $\mathfrak{gl}(n,\mathbb C)$},
     journal = {Sbornik. Mathematics},
     pages = {1375--1382},
     publisher = {mathdoc},
     volume = {191},
     number = {9},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_9_a5/}
}
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A. A. Tarasov. On some commutative subalgebras of the universal enveloping algebra of the Lie algebra $\mathfrak{gl}(n,\mathbb C)$. Sbornik. Mathematics, Tome 191 (2000) no. 9, pp. 1375-1382. http://geodesic.mathdoc.fr/item/SM_2000_191_9_a5/