Integrable models for shallow water with energy dependent spectral problems

We study the inverse problem for the so-called operators with energy depending potentials. In particular, we study spectral operators with quadratic dependence on the spectral parameter. The corresponding hierarchy of integrable equations includes the Kaup-Boussinesq equation. We formulate the inverse problem as a Riemann-Hilbert problem with a 1₂ reduction group. The soliton solutions are explicitly obtained.