We explore the detailed topological structure of the phase-space of the recently discovered three-dimensional integrable but nonseparable Hamiltonian system with velocity dependent potential. Two three-parameter families of three-dimensional tori which foliate the phase-space are identified. The complete classification, according to their topology, of the level sets corresponding to the critical points of the energy-momentum map, is accomplished. The relationship of the three-dimensional integrable system with important physical systems of current interest, namely, the hydrogen atoms in circularly polarized (CP) fields, in crossed magnetic and electric fields and in crossed magnetic and CP fields, is discussed.