In Part I[J. Math. Phys. 40, 1595 (1999)] we studied eigenfunctions of the quantum dynamics that defines the two-particle relativistic Calogero-Moser system with elliptic interaction. In the present paper we consider the same system with hyperbolic and trigonometric interactions. In these special regimes the eigenfunctions are shown to admit an elementary representation that is far more explicit than the "zero representation" of Part I. In particular, the new representation can be exploited to prove that the hyperbolic eigenfunctions can be chosen to be symmetric under interchanging position and momentum variables (self-duality). In the trigonometric case duality properties are derived, too, and several orthogonality and completeness results are obtained.
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