GENERALIZED LAX PAIRS, THE MODIFIED CLASSICAL YANG-BAXTER EQUATION, AND AFFINE GEOMETRY OF LIE-GROUPS

We generalize the usual Lax equation (GRAPHICS) where rho is an arbitrary representation of a Lie algebra g (the values of M) in a representation space V (the values of L). The usual classical r-matrix programme for Hamiltonian integrable systems is generalized to r-matrices taking values in g X V. The r-matrices are then considered as left invariant torsion-free covariant derivatives on a Lie group K (with Lie albegra V*). The Classical Yang-Baxter Equation (CYBE) is equivalent to the flatness of K whereas the Modified CYBE implies that K is an affine locally symmetric space. An example is discussed