[Python] How to find points that preserve given nearest neighbor distance?

I have a set of two-dimensional points, X with dimension of n by 2, and nearest neighbor distance of Dx (n by 1) where DX(i) being the distance of X(i) to its nearest neighbor. I want to generate nontrivial Y with the constraint that Dy=Dx. With nontrivial I mean Y can't be translation, rotation of reflection of X or any combination of these operations.

I tried to generate Y points iteratively with some random initial point and adding the subsequent points with the specified distance such that the new point doesn't change the previous points nearest neighbor. But then I'd need to sort distances and start with the highest distance and proceed to the lowest distance because otherwise if the next distance is higher, the nearest neighbor of the last point would be the previous point not the newly added point. With sorting the distances, I can get the desired points, but I prefer if they are more randomized. With distances sorted, all initial points are far from each other and all last points are close to each other.

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