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 prove the conjecture for filtrations of compact Lie algebras \(\mathfrak{g}\): the system is integrable in a noncommutative sense by means of polynomial integrals. Various constructions of complete commutative polynomial integrals for the system are also given.

1983 年 Bogoyavlenski 推测，如果李代数\(\mathfrak{g}_{0}\)上的欧拉方程是可积的，那么它们对半单李代数\(\mathfrak{g}\) 的某些扩展与李代数的过滤 \(\mathfrak{g}_{0}\subset\mathfrak{g}_{1}\subset\mathfrak{g}_{2}\dots\subset\mathfrak{g}_{n- 1}\subset\mathfrak{g}_{n}=\mathfrak{g}\)也是可积的。特别是，通过采用\(\mathfrak{g}_{0}=\{0\}\)和\({\mathfrak{so}}(n)\)和\(\mathfrak{u} (n)\)，我们有 Gel'fand – Cetlin 可积系统。我们证明了紧李代数\(\mathfrak{g}\)的过滤猜想：通过多项式积分，系统在非交换意义上是可积的。还给出了系统完全交换多项式积分的各种构造。