*This is one of those **"what is an $X$?"** questions so let me apologize in advance*.


By now I have already encountered the phrase "integrable hierarchy" in mathematical contexts (in particular the so called "Kdv hierarchy" which is apparently related to enumerative geometry of curves in some ways which are a total mystery to me) enough times to care about the meaning of these words. 


However when i type these words in google most of the results are links to physics papers and I haven't been able to find anything clarifying about the meaning of this phrase, let alone a precise mathematical definition. Hence the question:



> 
> What is an integrable hierarchy? Where do they come from? What are their applications?
> 
> 
> 


**EDIT:** Let me emphasize that this question is mainly about the "hierarchy" part. I understand (at least on a basic level) what is an integrable system and I know of several good mathematical references for this topic. The emphasis here is on what constitutes an **integrable hierarchy** as apposed to a plain old "integrable system".



What is an "integrable hierarchy"? (to a mathematician)
An integrable hierarchy is another name for a system of commuting Hamiltonian flows. The word "hierarchy" is used because a countably infinite number of commuting flows is obtained recursively.  

[For the definition of a commuting flow, see for example the first part of this [MO question.](https://mathoverflow.net/questions/98534/when-do-commuting-hamiltonian-flows-have-commuting-generators)] 


They arise from [integrals of motion](https://en.wikipedia.org/wiki/Constant_of_motion#Integral_of_motion) which are in involution (meaning that the [Poisson bracket](https://en.wikipedia.org/wiki/Poisson_bracket) of any pair vanishes). 


Commuting flows are useful, because they can be solved by the [inverse scattering transform](https://en.wikipedia.org/wiki/Inverse_scattering_transform) technique.


For an introduction from a mathematical perspective, see for example [Introduction to integrable systems: open Toda lattice, KP, and KdV-hierarchies.](http://gokovagt.org/proceedings/2009/ggt09-shapiro.pdf)


