The argument shift method and maximal commutative subalgebras of Poisson algebras (a joint work with O. Yakimova)

Let $q$ be an algebraic Lie algebra and $S(q)$ the corresponding symmetric algebra equipped with the standard Poisson–Lie structure. Set
$$ b(q)=(\dim q+\mathrm{ind}q)/2. $$
As is well-known, the transcendence degree of any Poisson commutative subalgebra of $S(q)$ does not exceed $b(q)$. In 1978, Mischenko and Fomenko proposed a method for constructing commutative subalgebras of maximal transendence degree (the so-called “argument shift method”). However, for arbitrary $q$, these Mischenko–Fomenko subalgebras are not necessarily maximal with respect to inclusion.
I'll describe general sufficient conditions on $q$ that guarantee us the maximality of M-F subalgebras. These conditions are satisfied if $q$ is semisimple and in some other interesting cases.
The proof is based on (a generalisation of) Bolsinov' criterion.