Energy-dependent potentials revisited: A universal hierarchy of hydrodynamic type

A hierarchy of infinite-dimensional systems of hydrodynamic type is considered and a general scheme for classifying its reductions is provided. Wide families of integrable systems including, in particular, those associated with energy-dependent spectral problems of Schr\" odinger type, are characterized as reductions of this hierarchy. $N$-phase type reductions and their corresponding Dubrovin equations are analyzed. A symmetry transformation connecting different classes of reductions is formulated.