For a rational differential operator \({L=AB^{-1}}\), the Lenard–Magri scheme of integrability is a sequence of functions \({F_n, n \geq 0}\), such that (1) \({B(F_{n+1})=A(F_n)}\) for all \({n \geq 0}\) and (2) the functions \({B(F_n)}\) pairwise commute. We show that, assuming that property (1) holds and that the set of differential orders of \({B(F_n)}\) is unbounded, property (2) holds if and only if L belongs to a class of rational operators that we call integrable. If we assume moreover that the rational operator L is weakly non-local and preserves a certain splitting of the algebra of functions into even and odd parts, we show that one can always find such a sequence (F _n ) starting from any function in Ker B. This result gives some insight in the mechanism of recursion operators, which encode the hierarchies of the corresponding integrable equations.

对于有理微分算子\({L=AB^{-1}}\)，Lenard–Magri 可积方案是一系列函数\({F_n, n \geq 0}\)，使得 (1) \({B(F_{n+1})=A(F_n)}\)对于所有\({n \geq 0}\)和 (2) 函数\({B(F_n)}\)成对通勤。我们证明，假设性质（1）成立并且\({B(F_n)}\)的微分阶数集是无界的，性质（2）成立当且仅当L属于一类有理算子，我们称之为可积的。此外，如果我们假设有理算子L是弱非局部的，并且保留了函数代数分成偶数和奇数部分的某种分裂，我们表明人们总是可以从 Ker 中的任何函数开始找到这样一个序列 ( F _n ) B.​这个结果让我们对递归算子的机制有了一些了解，递归算子对相应的可积方程的层次结构进行编码。