  In 1983 Bogoyavlenski conjectured that if the Euler equations on a Lie
algebra $\mathfrak g_0$ are integrable, then their certain extensions to
semisimple lie algebras $\mathfrak g$ related to the filtrations of Lie
algebras $\mathfrak g_0\subset \mathfrak g_1\subset \mathfrak
g_2\dots\subset\mathfrak g_{n-1}\subset \mathfrak g_n=\mathfrak g$ are
integrable as well. In particular, by taking $\mathfrak g_0=\{0\}$ and natural
filtrations of $\mathfrak{so}(n)$ and $\mathfrak{u}(n)$, we have
Gel'fand-Cetlin integrable systems. We proved the conjecture for filtrations of
compact Lie algebras $\mathfrak g$: the system are integrable in a
noncommutative sense by means of polynomial integrals. Various constructions of
complete commutative polynomial integrals for the system are also given.
