Integrable matrix equations related to pairs of compatible associative   algebras
  We study associative multiplications in semi-simple associative algebras over C compatible with the usual one. An interesting class of such multiplications is related to the affine Dynkin diagrams of A, D, E-type. In this paper we investigate in details the multiplications of the A-type and integrable matrix ODEs and PDEs generated by them. 
Alexander Odesskii and Vladimir Sokolov