On new constructions in the Blaschke-Bol problem

We find several essentially new constructions of hexagonal $3$-webs based on a combination of quadratic and linear families of circles. They are used to construct $5$ new types of hexagonal $3$-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 $3$-webs and their properties. In conclusion, we formulate several conjectures and open problems.
Bibliography: 13 titles.