  We give a geometric interpretation of all the $m$-th elliptic integrable
systems associated to a $k'$-symmetric space $N=G/G_0$ (in the sense of C.L.
Terng). It turns out that we have to introduce the integer $m_{k'}$ defined by
m_{1}=0 and m_{k'}= [(k'+1)/2]. Then the general problem splits into three
cases : the primitive case ($m < m_{k'}$), the determined case ($m_{k'}\leq m
\leq k'-1$) and the underdetermined case ($m \geq k'$). We prove that we have
an interpretation in terms of a sigma model with a Wess-Zumino term. Moreover
we prove that we have a geometric interpretation in terms of twistors. See the
abstract in the paper for more precisions.
