A generalized Melvin solution for an arbitrary simple finite-dimensional Lie algebra G is considered. The solution contains a metric, n Abelian 2-forms and n scalar fields, where n is the rank of G. It is governed by a set of n moduli functions Hs(z) obeying n ordinary differential equations with certain boundary conditions imposed. It was conjectured earlier that these functions should be polynomials—the so-called fluxbrane polynomials. These polynomials depend upon integration constants qs, s= 1 , ⋯ , n. In the case when the conjecture on the polynomial structure for the Lie algebra G is satisfied, it is proved that 2-form flux integrals Φ s over a proper 2d submanifold are finite and obey the relations qsΦ s= 4 πnshs, where the hs> 0 are certain constants (related to dilatonic coupling vectors) and the ns are powers of the polynomials, which are components of a twice dual Weyl vector in the basis of simple (co-)roots, s= 1 , ⋯ , n. The main relations of the paper are valid for a solution corresponding to a finite-dimensional semi-simple Lie algebra G. Examples of polynomials and fluxes for the Lie algebras A1, A2, A3, C2, G2 and A1+ A1 are presented. © 2017, The Author(s).