  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 B"acklund transformation for
circle patterns is studied. Further, a class of isomonodromic solutions of the
aforementioned integrable system is introduced, including circle patterns
analogons to the analytic functions $z^\alpha$ and $\log z$.
