  A generalized Melvin solution for an arbitrary simple finite-dimensional Lie
algebra $\cal G$ is considered. The solution contains a metric, $n$ Abelian
2-forms and $n$ scalar fields, where $n$ is the rank of $\cal G$. 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 polynomials - the 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 $\cal G$ 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 $h_s > 0$ are certain constants
(related to dilatonic coupling vectors) and $n_s$ 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
$\cal G$. 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.
