Algebraic $2$-valued group structures on $\mathbb P^1$, Kontsevich-type polynomials, and multiplication formulas. I

The theory of a $2$-valued algebraic group structure on a complex plane and complex projective line is developed. In this theory, depending on the choice of the neutral element, the local multiplication law is given by the Buchstaber polynomial or the generalized Kontsevich polynomial. One of the most exciting results of the first part of our work is a simple construction of a $2$-valued algebraic group structure on $\mathbb C$ different from well known coset-construction.