It seems like people talk about many different versions of the problem, but only certain enthusiasts take any one version to an extreme depth. I'll be the enthusiast who goes deep on the "sum and product, lower bound 3, no upper bound" version, then!

I have looked at the van Ditmarsch papers, though not though all the references they list. They use a similar algorithm to the one I give here to list solutions, though there is no talk of the generalization to arbitrary upper bound. The only claim I have seen regarding classifying all pairs of solutions was the one made by Born-Hurkens-Woeginger conjecturing there were infinitely many (a conjecture which I feel I am near proving false)