Abstract
We investigate the open question asking whether there exist independent systems of three equations over three unknowns admitting non-periodic solutions, formulated in 1983 by Culik II and Karhumäki. In particular, we give a negative answer to this question for a large class of systems. More specifically, the question remains open only for a well specified class of systems. We also investigate systems of two equations over three unknowns for which we give necessary and sufficient conditions for admitting at most quasi-periodic solutions, i.e., solutions where the images of two unknowns are powers of a common word. In doing so, we also give a number of examples showing that these conditions represent a boundary point between systems admitting purely non-periodic solutions and those admitting at most quasi-periodic ones.
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