New Hamiltonization of the Schroedinger equation by corresponding nonlinear equation for the potential

A new nonlinear equation for potential, which, with corresponding Schroedinger equation, composes a new Hamiltonian system is found. The set of solutions of this extended quantum theory contains conformal quantum mechanics. Multidimensional many particle and supersymmetric generalizations are given. For inverse-square potentials with arbitrary angular dependence we find an extra integral of motion and the radial dynamics is integrated in quadratures. Nambu-theoretical formulation of the extended quantum theory is constructed