In this the sequel to arXiv1910.05816, we derive a necessary and sufficient condition characterizing which real-valued continuous solutions of a multivariate Goldie functional equation express homomorphy between the multivariate Popa groups defined and characterized in the earlier work. This enables us to deduce that all (real-valued) continuous solutions are homomorphisms between such groups. We use this result also to characterize as Popa homomorphisms smooth solutions of a related more general equation, also of Levi-Civita type. A key result here (Theorem 2) on purely radial behaviour is generalized in arXiv2105.07794 to a Banach-algebra setting involving radial tilting behaviour.
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