  The survey is devoted to algebraic structures related to integrable ODEs and
evolution PDEs. A description of Lax representations is given in terms of
vector space decomposition of loop algebras into a direct sum of Taylor series
and a complementary subalgebra. Examples of complementary subalgebras and
corresponding integrable models are presented. In the framework of the
bi-Hamiltonian approach compatible associative algebras related affine Dynkin
diagrams are considered. A bi-Hamiltonian origin of the classical elliptic
Calogero-Moser models is revealed.
