Geometric Interpretation of Second Elliptic Integrable System

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 $G/G_0 \hookrightarrow \Sigma(G/H)$. We begin the paper by an example: $G/H=\R^4$. We study also the structure of 4-symmetric bundles over Riemannian symmetric spaces.