  The Mischenko-Fomenko argument shift method allows to construct commutative
subalgebras in the symmetric algebra $S(\mathfrak g)$ of a finite-dimensional
Lie algebra $\mathfrak g$. For a wide class of Lie algebras, these commutative
subalgebras appear to be complete, i.e. they have maximal transcendence degree.
However, for many algebras, Mischenko-Fomenko subalgebras are incomplete or
even empty. In this case, we suggest a natural way how to extend
Mischenko-Fomenko subalgebras, and give a completeness criterion for these
extended subalgebras.
