Hydrodynamic reductions of multi-dimensional dispersionless PDEs: the test for integrability

A (d+1)-dimensional dispersionless PDE is said to be integrable if its n-component hydrodynamic reductions are locally parametrized by (d-1)n arbitrary functions of one variable. Given a PDE which does not pass the integrability test, the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which, when added to the equation, make it an integrable system in fewer dimensions (if consistent).