Limits of integrable Hamiltonians on semisimple Lie algebras

Let $\mathfrak{g}$ be a semisimple Lie algebra, and let $P(\mathfrak{g})$ be the corresponding Poisson algebra. With each regular element $a\in \mathfrak{g}$, the argument shift method associates a commutative subalgebra $F(a)\subset P(\mathfrak{g})$, whose transcendence degree is maximal possible, i.e., is equal to the dimension of a Borel subalgebra of $\mathfrak{g}$. When a tends to a singular element in a proper way, the subalgebra $F(a)$ tends to some commutative subalgebra of the same transcendence degree. The cases when a tends to a singular element remaining in the same Cartan subalgebra, were investigated in old works of the speaker (1990) and V.V. Shuvalov (2002). Some other cases will be discussed in the talk. An interesting problem is to describe the variety of integrable quadratic Hamiltonians arising in this way.