Is it possible to become a research mathematician without lecturing duties? I couldn't find any information on this.
The answer to your question is yes: there are research-only career pathways for mathematicians. One way of achieving this is to secure a job in a research-only institution, e.g. Max-Planck Institute in Germany, Academy of Sciences in Russia and other post-Soviet countries, or one of many government research labs / institutes in the USA.
Looking for a research-only career in a University can be much more challenging, if at all possible. This would usually imply buying off your teaching duties from a sequence of successful grants, and supervising many PhD students (which also involves a fair amount of teaching).
I can also recommend this article.