  In this part we show that the diophantine equation
$X^5+Y^5=\varepsilon^5(1-X^5Y^5)$, where $\varepsilon=\frac{-1+\sqrt{5}}{2}$,
has solutions in specific abelian extensions of quadratic fields
$K=\mathbb{Q}(\sqrt{-d})$ in which $-d \equiv \pm 1$ (mod $5$). The coordinates
of these solutions are values of the Rogers-Ramanujan continued fraction
$r(\tau)$, and are shown to be periodic points of an algebraic function.
