Summary: We present a novel approach to analyze uniqueness in nonlinear inverse problems, using a novel bifocal Newtonian algorithm for identifying pairs of non-unique solutions for any potential data set, prior to any data collection. For the case when the shape of the forward function depends on control parameters that can be tuned to reduce non-uniqueness, we present a second algorithm which minimizes the sum of squared distances between each pair of non-unique solutions. Both algorithms are also relevant in the presence of uncertainty, which we demonstrate by applying them to a simple nonlinear location problem.