A generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra mathcal G is considered. The solution contains a metric, n Abelian 2-forms and n scalar fields, where n is the rank of mathcal G. It is governed by a set of n moduli functions H_s(z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomialsâthe so-called fluxbrane polynomials. These polynomials depend upon integration constants q_s, s = 1,dots ,n. In the case when the conjecture on the polynomial structure for the Lie algebra mathcal G is satisfied, it is proved that 2-form flux integrals Phi ^s over a proper 2d submanifold are finite and obey the relations q_s Phi ^s = 4 pi n_s h_s, where the h_s > 0 are certain constants (related to dilatonic coupling vectors) and the n_s are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, s = 1,dots ,n. The main relations of the paper are valid for a solution corresponding to a finite-dimensional semi-simple Lie algebra mathcal G. Examples of polynomials and fluxes for the Lie algebras A_1, A_2, A_3, C_2, G_2 and A_1 + A_1 are presented.