The goal of this crash course is to make a brief introduction into the beautiful world of integrable hierarchies. We do not intend to give a general survey of integrable system theory but rather want to use few known examples to introduce notions and initial circle of ideas specific for integrable equations. We used [A] as a reference for the first section, [P] for the second section, and [MJD]...

A class of n-ary Poisson structures of constant rank is indicated. Then, one proves that the ternary Poisson brackets are exactly those which are deened by a decomposable 3-vector eld. The key point is the proof of a lemma which tells that an n-vector (n 3) is decomposable ii all its contractions with up to n ? 2 covectors are decomposable. In the last years, several authors have studied genera...