What are the non-aqueous solvents for sodium polyacrylate?
Sodium polyacrylate can locally organize solvent molecules to become soluble. Since the backbone is huge compared to water, the water aligns with its more positively charged end towards the carboxylate site while the sodium ions end up on the negative end of the water. I can't find the good papers right now, but if you search google scholar for using polyacrylic acid as a draw solute for forward osmosis you can dig way deeper into the mechanisms.

What this means practically is that you need really polar solvents that can line up with the ions. Wikipedia has a good page explaining solvent classification in more detail with a table of solvents including their dipole moment (quantifying the polarity). As suggested by Mithoron, DMSO seems to be a good option. Unfortunately, after that you start getting into much nastier solvents like acetonitrile and DMF.