  Motivated by the study of certain nonlinear wave equations (in particular,
the Camassa-Holm equation), we introduce a new class of generalized indefinite
strings associated with differential equations of the form
\[-u"=z\,u\,\omega+z^2u\,\upsilon\] on an interval $[0,L)$, where $\omega$ is a
real-valued distribution in $H^{-1}_{\mathrm{loc}}[0,L)$, $\upsilon$ is a
non-negative Borel measure on $[0,L)$ and $z$ is a complex spectral parameter.
Apart from developing basic spectral theory for these kinds of spectral
problems, our main result is an indefinite analogue of M. G. Krein's celebrated
solution of the inverse spectral problem for inhomogeneous vibrating strings.
