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We study commutative subalgebras in the symmetric algebra $S(\mathfrak{g})$of a finite-dimensional Lie algebra $\mathfrak{g}$. A. M. Izosimov introducedextended Mischenko-Fomenko subalgebras $\tilde{\mathcal{F}}_a$ and gave acompleteness criterion for them. We generalize his construction and extendMischenko-Fomenko subalgebras with the shifts of all semi-invariants of$\mathfrak{g}$. We prove that the new commutative subalgebras have the sametranscendence degree as $\tilde{\mathcal{F}}_a$.

Shifts of semi-invariants and complete commutative subalgebras in polynomial Poisson algebras