  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.
