Classification of regular circle three-webs up to circular transformations

A curvilinear three-web formed by three pencils of circles is called a circle web. Generally speaking, the circle three-web is not regular, i.e., it is not locally diffeomorphic to a web formed by three families of parallel straight lines. In this paper, all regular circle three-webs are classified up to circular transformations. The main result is as follows: there exist 48 nonequivalent (with respect to circular transformations) types of regular three-webs. Five of them contain $\infty^3$ nonequivalent webs each, 11 types contain $\infty^2$ nonequivalent webs each, 12 types contain $\infty^1$ nonequivalent webs each; 5 webs admit a one-parameter group of automorphisms.