On generalized Melvin solutions for Lie algebras of rank 4

We deal with generalized Melvin-like solutions associated with Lie algebras of rank $4$ ($A_4$, $B_4$, $C_4$, $D_4$, $F_4$). Any solution has static cylindrically-symmetric metric in $D$ dimensions in presence of four Abelian 2-forms and four scalar fields. The solution is governed by four moduli functions $H_s(z)$ ($s = 1,...,4$) of squared radial coordinate $z=\rho^2$ obeying four differential equations of the Toda chain type. These functions are polynomials of powers $(n_1,n_2, n_3, n_4) = (4,6,6,4), (8,14,18,10), (7,12,15,16), (6,10,6,6), (22,42,30,16)$ for Lie algebras $A_4$, $B_4$, $C_4$, $D_4$, $F_4$, respectively. The asymptotic behaviour for the polynomials at large $z$ is governed by an integer-valued $4 \times 4$ matrix $\nu$ connected in a certain way with the inverse Cartan matrix of the Lie algebra and (in $A_4$ case) the matrix representing a generator of the $\mathbb{Z}_2$-group of symmetry of the Dynkin diagram. The symmetry properties and duality identities for polynomials are studied. We also present 2-form flux integrals over a $2$-dimensional submanifold. Dilatonic black hole analogs of the obtained Melvin-type solutions, e.g. "fantom" ones, are also considered. The phantom black holes are described by fluxbrane polynomials under consideration.