Mechanical systems related to the Schrödinger spectral problem

We present a variety of integrable mechanical systems which are embedded into the KdV and the Harry Dym hierarchies of soliton equations as their restricted flows. Integrable structures of these systems are systematically derived from the properties of the underlying hierarchies of equations. It is illustrated by the key example of the Garnier system which describes the motion of a particle in a quartic ‘wine bottle’ potential.