The goal of this study was to train an artificial neural network to generate accurate saccades in Listing's plane and then determine how the hidden units performed the visuomotor transformation. A three-layer neural network was successfully trained, using back-prop, to take in oculocentric retinal error vectors and three-dimensional eye orientation and to generate the correct head-centric motor error vector within Listing's plane. Analysis of the hidden layer of trained networks showed that explicit representations of desired target direction and eye orientation were not employed. Instead, the hidden-layer units consistently divided themselves into four parallel modules: a dominant "vector-propagation" class (approximately 50% of units) with similar visual and motor tuning but negligible position sensitivity and three classes with specific spatial relations between position, visual, and motor tuning. Surprisingly, the vector-propagation units, and only these, formed a highly precise and consistent orthogonal coordinate system aligned with Listing's plane. Selective "lesions" confirmed that the vector-propagation module provided the main drive for saccade magnitude and direction, whereas a balance between activity in the other modules was required for the correct eye-position modulation. Thus, contrary to popular expectation, error-driven learning in itself was sufficient to produce a "neural" algorithm with discrete functional modules and explicit coordinate systems, much like those observed in the real saccade generator.