Surfaces containing several circles through each point (on a joint work with R. Krasauskas)

Motivated by potential applications in architecture, we study surfaces in 3-dimensional Euclidean space containing several circles through each point. Finding all such surfaces is a challenging open problem. We provide some bright examples and reduce the problem to a nice algebraic problem of finding Pythagorean n-tuples of polynomials.
Our main tools is a generalization of the Schicho theorem on the parametrization of the surfaces containing two conics through each point. We are going to state and prove several lemmas to the theorem.
A substantial part of the talk is elementary and is accessible even for high school students. Several open problems are stated.