Geometric interpretation of second elliptic integrable systems

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/G0G/G0 can be embedded into the twistor space of the corresponding symmetric space G/HG/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/G0↪Σ(G/H)G/G0↪Σ(G/H). We begin the paper with an example: G/H=R4G/H=R4. We also study the structure of 4-symmetric bundles over Riemannian symmetric spaces.