Invariant algebraic surfaces of the Lorenz system

In this paper we provide a complete classification of the Darboux invariants, of the irreducible Darboux polynomials, of the rational first integrals and of the algebraic integrability for the classical Lorenz system ẋ=s(y−x), ẏ=rx−y−xz, ż=−bz+xy. In the proofs, we use the weight homogeneous polynomials and the method of characteristic curves for solving linear partial differential equations.