Remarks on the complete integrability of dynamical systems with fermionic variables

Abstract

The authors study the role of (1,1) graded tensor field T in the analysis of complete integrability of dynamical systems with fermionic variables. They 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. They clarify this fact by constructing an odd tensor for two examples, a supersymmetric Toda chain and a supersymmetric harmonic oscillator. They explicitly show that it cannot be a recursion operator since it does not allow new constants of motion to be built from the first two, in contrast to what usually happens with ordinary, i.e. nongraded systems.