Landi G., Marmo G., Vilasi G.
REMARKS ON THE COMPLETE INTEGRABILITY OF DYNAMICAL SYSTEMS WITH 
FERMIONIC VARIABLES
(34K, LaTeX)

ABSTRACT.  We study the r\^{o}le of $(1,1)$ graded tensor field 
$T$ in the analysis of complete integrability of dynamical systems 
with fermionic variables. We find that such a tensor $T$ can be a 
recursion operator if and only if $T$ is even as a graded map, 
namely, if and only if $p(T) = 0$. 
We clarify this fact by
constructing an odd tensor for two examples, a supersymmetric Toda 
chain and a supersymmetric harmonic oscillator. We explicitly show 
that it cannot be a recursion operator not allowing then to build new 
constants of the motion out of the first two ones in contrast to what
usually 
happens with ordinary, i.e. non graded systems.