Solution of the three-dimensional inverse problem

The three dimensional inverse problem for a material point of unit mass, moving in an autonomous conservative field, is solved. Given a two-parametric family of space curvesf(x, y, z)=c ₁,g(x, y, z)=c ₂, it is shown that, in general, no potentialU=U(x, y, z) exists which can give rise to this family. However, if the given functionsf(x, y, z) andg(x, y, z) satisfy certain conditions, the corresponding potentialU(x, y, z), as well as the total energyE=E(f, g) are determined uniquely, apart from a multiplicative and an additive constant.