Mathematics

We will discuss sufficient and necessary conditions for a rational differential operator L=AB^{-1} to generate a Lenard-Magri scheme of integrability. Provided that L can be applied infinitely many times to a root function, the functions in the hierarchy pairwise commute if and only if L lies in a class
of rational operators that we call integrable. If we assume moreover that L is weakly non-local and preserves a Z/2Z-grading of the algebra of functions, we show that one can always construct an integrable hierarchy starting from any function in Ker B.