Hexagonal circle patterns and integrable systems: patterns with the multi-ratio property and Lax equations on the regular triangular lattice

Hexagonal circle patterns are introduced, and a subclass thereof is studied in detail. It is characterized by the following property: for every circle the multi-ratio of its six intersection points with neighboring circles is equal to −1. The relation of such patterns with an integrable system on the regular triangular lattice is established. A kind of a Bäcklund transformation for circle patterns is studied. Further, a class of isomonodromic solutions of the aforementioned integrable system is introduced, including circle patterns analogs to the analytic functions z
^α
and logz.