In this paper we present the second part of a study of spherically-symmetric solutions of the Schrödinger-Newton equations for a single particle (Penrose R 1998 Quantum computation, entanglement and state reduction Phil. Trans. R. Soc. 356 1-13). We show that there exists an infinite family of normalizable, finite energy solutions which are characterized by being smooth and bounded for all values of the radial coordinate. We therefore provide analytical support for our earlier numerical integrations (Moroz I M et al 1998 Spherically-symmetric solutions of the Schrödinger-Newton equations Classical and Quantum Gravity 1998 15 2733-42).