The algebraic version of the Birkhoff conjecture is solved completely for billiards with a piecewise C2-smooth boundary on surfaces of constant curvature: Euclidean plane, sphere, and Lobachevsky plane. Namely, we obtain a complete classification of billiards for which the billiard geodesic flow has a nontrivial first integral depending polynomially on the velocity. According to this classification, every polynomially integrable convex bounded planar billiard with C2-smooth boundary is an ellipse. This is a joint result of M. Bialy, A.E. Mironov, and the author. The proof consists of two parts. The first part was given by Bialy and Mironov in their two joint papers, where the result was reduced to an algebraic-geometric problem, which was partially studied there. The second part is the complete solution of the algebraic-geometric problem presented below.