On new constructions in the Blaschke-Bol problem

We find several essentially new constructions of hexagonal -webs based on a combination of quadratic and linear families of circles. They are used to construct new types of hexagonal -webs, which is an advance in the solution of the Blaschke-Bol problem (1938) on the classification of such webs. Unlike many known examples, in our proofs we give an explicit parallelizing diffeomorphism. We give a brief survey of all known examples of hexagonal -webs and their properties. In conclusion, we formulate several conjectures and open problems. Bibliography: 13 titles.