An inverse spectral problem for the Sturm-Liouville operator with a singular potential from class $W_2^{-1}$ is solved by method of mappings. We prove uniqueness theorem, develop constructive algorithm solution and obtain necessary sufficient conditions solvability in self-adjoint non-self-adjoint cases.

We discuss direct and inverse spectral theory for a Sturm–Liouville type problem with a quadratic dependence on the eigenvalue parameter, −f ′′ + 1 4 f = z ωf + zυf, which arises as the isospectral problem for the conservative Camassa–Holm flow. In order to be able to treat rather irregular coefficients (that is, when ω is a real-valued Borel measure on R and υ is a non-negative Borel measure o...