What different routes are there to publish research besides academia? I would love to work on publications but it is not practical for me to return to an institution right now.
I don't know what your current industry is, but Google publishes their research and collaborates with academic institutions is one that immediately comes to mind. The government (NASA, NSA for one), tech companies (IBM, Microsoft, etc.), medical companies, aerospace/defense (JPL), just to name a few. I am sure there are way more than I could think of and I am sure others will care to fill in as well.
Research is not exclusive to academia.
I've never seen a math journal that requires academic credentials or affiliation to submit a paper, and I've published several math papers without an academic affiliation. You can put your employer as your affiliation or even "Independent Researcher". The hard part is writing the papers themselves. Getting a paper in a decent math journal as an outsider is rare, not because the journals ban or are biased against outsiders, but just because it's rare for an outsider to write a decent math paper.