Deformations of dispersionless KdV hierarchies

The obstructions to the existence of a hierarchy of hydrodynamic conservation laws are studied for a multicomponent dispersionless KdV system. It is proved that if the lowest order obstruction vanishes then all higher obstructions automatically vanish, if and only the underlying algebra is a Jordan algebra. Deformations of these multicomponent dispersionless KdV-type equations are also studied. It is shown that no new obstructions appear and, hence, that the existence of a fully deformed hierarchy depends only on the existence of a single purely hydrodynamic conservation law.