  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.
