Bäcklund transformations for the one-dimensional Schrödinger equation

We study the system of equations which bases on the one-dimensional Schrödinger equation and connects the potential, amplitude, and phase functions. Using the methods of compatibility theory of systems of partial differential equations, we obtain the completely integrable systems that connect only two functions of the above three. As a corollary, we construct some exact solutions of the Schrödinger equation.