The article considered the Kaup-Boussinesq type system. It is shown that the system is a fully integrable system. Exactly, the time evolution of the scattering data for a quadratic pencil of Sturm-Liouville operator is founded. The obtained equalities completely determine the scattering data for any t, which makes it possible to apply the method of the inverse scattering problem to solve the Cauchy problem for the Kaup-Boussinesq type system.