Chopping integrals of the full symmetric Toda system, a new approach

In my talk I will try to answer the questions that has been causing my anxiety for a rather long time: where do the additional integrals of the full symmetric Toda system come from, why they are rational and what does all this have to do with "chopping". Even if we can use the AKS method there remains the question, why do the initial functions actually commute (and whether it is possible to find other with the same property). The known answers were concerned either with rather hard straightforward computations, or with the properties of a Gaudin system; they look pretty complicated. In my talk I will show how one can obtain these integrals with the help of some simple differential operators (in the manner of the argument shift method). Besides this, we will discuss some other possible integrals as well as the method to solve the corresponding flows by QR decomposition.
The talk is based on a common work with Yu. Chernyakov and D. Talalaev.