Geometric interpretation of second elliptic integrable systems

Abstract

In this paper we give a geometrical interpretation of all the second elliptic integrable systems associated to 4-symmetric spaces. We first show that a 4-symmetric space G / G 0 can be embedded into the twistor space of the corresponding symmetric space G / H . Then we prove that the second elliptic system is equivalent to the vertical harmonicity of an admissible twistor lift J taking values in G / G 0 ↪ Σ ( G / H ) . We begin the paper with an example: G / H = R 4 . We also study the structure of 4-symmetric bundles over Riemannian symmetric spaces.