The KdV hierarchy is a family of evolutions on a Schrödinger operator that preserves its spectrum. Canonical systems are a generalization of Schrödinger operators, that nevertheless share many features with Schrödinger operators. Since this is a very natural generalization, one would expect that it would also be straightforward to build a hierarchy of isospectral evolutions on canonical systems analogous to the KdV hierarchy. Surprisingly, we show that there are many obstructions to constructing a hierarchy of flows on canonical systems that obeys the standard assumptions of the KdV hierarchy. This suggests that we need a more sophisticated approach to develop such a hierarchy, if it is indeed possible to do so.

KdV 层次结构是薛定谔算子上的一系列演化，保留了其频谱。规范系统是薛定谔算子的推广，但与薛定谔算子共享许多特征。由于这是一个非常自然的概括，人们会期望在类似于 KdV 层次结构的典型系统上构建等谱演化层次结构也很简单。令人惊讶的是，我们表明在遵循 KdV 层次结构的标准假设的规范系统上构建流层次结构存在许多障碍。这表明我们需要一种更复杂的方法来开发这样的层次结构，如果确实有可能的话。