Generalized Lax pairs,the modified classical Yang-Baxter equation,and affine geometry of Lie groups

We generalize the usual Lax equationd/dt L=[M, L] byd/dt L=–(M)L, where is an arbitrary representation of a Lie algebra g (the values ofM) in a representation spaceV (the values ofL). The usual classicalr-matrix programme for Hamiltonian integrable systems is generalized tor-matrices taking values in gV. Ther-matrices are then considered as left invariant torsion-free covariant derivatives on a Lie groupK (with Lie algebraV ^*). The Classical Yang-Baxter Equation (CYBE) is equivalent to the flatness ofK whereas the Modified CYBE implies thatK is an affine locally symmetric space. An example is discussed.     

Krichever-Novikov algebras for more than two points

Krichever-Novikov algebras of meromorphic vector fields with more than two poles on higher genus Riemann surfaces are introduced. The structure of these algebras and their induced modules of forms of weight is studied.     

Polynomial operator matrices as generators: The general case

Of concern are systems of linear evolution equations