  Generalized Melvin solutions for rank-$3$ Lie algebras $A_3$, $B_3$ and $C_3$
are considered. Any solution contains metric, three Abelian 2-forms and three
scalar fields. It is governed by three moduli functions $H_1(z),H_2(z),H_3(z)$
($z = \rho^2$ and $\rho$ is a radial variable), obeying three differential
equations with certain boundary conditions imposed. These functions are
polynomials with powers $(n_1,n_2, n_3) = (3,4,3), (6,10,6), (5,8,9)$ for Lie
algebras $A_3$, $B_3$, $C_3$, respectively. The solutions depend upon
integration constants $q_1, q_2, q_3 \neq 0$.
  The power-law asymptotic relations for polynomials at large $z$ are governed
by integer-valued $3 \times 3$ matrix $\nu$, which coincides with twice the
inverse Cartan matrix $2 A^{-1}$ for Lie algebras $B_3$ and $C_3$, while in the
$A_3$ case $\nu = A^{-1} (I + P)$, where $I$ is the identity matrix and $P$ is
a permutation matrix, corresponding to a generator of the $\mathbb{Z}_2$-group
of symmetry of the Dynkin diagram. The duality identities for polynomials and
asymptotic relations for solutions at large distances are obtained. 2-form flux
integrals over a $2$-dimensional disc of radius $R$ and corresponding Wilson
loop factors over a circle of radius $R$ are presented.
