We use the methods that were developed by Adler and van Moerbeke to determine explicit equations for a certain moduli space, that was studied by Narasimhan and Ramanan. Stated briefly it is, for a fixed non-hyperelliptic Riemann surface $\Gamma$ of genus $3$, the moduli space of semi-stable rank two bundles with trivial determinant on $\Gamma$. They showed that it can be realized as a projective variety, more precisely as a quartic hypersurface of ${\mathbb{P}}^7$, whose singular locus is the Kummer variety of $\Gamma$. We first construct...