Each choice of an arbitrary nonzero function f of the four immersion parameters is shown to determine 16N[f] distinguishable classes of two-parameter families of immersions of Einstein-Riemann spacetimes in six-dimensional flat spaces, where N[f] is the number of regular immersion parameter domains. The metric tensors, curvature tensors and the immersion loci are calculated in a closed form, and these calculations involve only finitely many algebraic operations. The presence of the arbitrary function provides the opportunity for study of the behaviour of multiple isolated singularities and/or 'shape' functions in general relativity.
