Algorithm 4.2 of the paper is implemented in Sage code here.
An html version of this
worksheet is available here.
This code determines completability, and in the 1 or 2 component
case, it returns an explicit completion.
When there are more than 2 components, the fiber is positive-dimensional, which means that to choose our completion, we instead optimize some function. The code in Maple to carry out this kind of computation with Lagrange techniques is here