The free undamped vibrations of rods, horns and taut strings are governed by second-order differential equations. It is known that the inverse problem, namely the reconstruction
of such a system, e.g. the reconstruction of the cross-sectional profile of a rod, requires the
knowledge of two free vibration spectra corresponding to two different sets of end conditions.
This paper is concerned with families of second-order systems which have one spectrum in
common. The analysis is based on the reduction of the governing equation to the standard
Sturm-Liouville form, the use of the Darboux lemma, and the research of Trubowitz, Poschel,
Deift and others. In particular the paper establishes necessary and sufficient conditions for
isospectral flow from one rod to another rod with the same end conditions, using double Darboux transformation.
