Integrable equations on $\mathbb Z$-graded Lie algebras

Evolution systems with $L$–$A$-pairs in $\mathbb Z$-graded Lie algebras are investigated. Some different hierarchies of integrable systems are associated with the same $L$-operator. They correspond to different decompositions of zero component of the $\mathbb Z$-graded algebra in a direct sum of two subalgebras. As the result, new examples of multi-component integrable systems are constructed.