An integrable hierarchy is another name for a system of commuting Hamiltonian flows.    
<sub>[For the definition of a commuting flow, see for example the first part of this <A HREF="https://mathoverflow.net/questions/98534/when-do-commuting-hamiltonian-flows-have-commuting-generators">MO question.</A>]</sub> 
   
They arise from integrals of motion which are in involution (meaning that the Poisson bracket of any pair vanishes).