On generalized Melvin solutions for Lie algebras of rank 4

We deal with generalized Melvin-like solutions associated 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 the presence four Abelian two-form and scalar fields. The is governed by moduli functions $$H_s(z)$$ $$s = 1,\ldots ,4$$ ) squared radial coordinate $$z=\rho ^2$$ obeying differential equations Toda chain type. These are polynomials 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 respectively. asymptotic behaviour at large z an integer-valued $$4 \times 4$$ matrix $$\nu $$ connected a certain way inverse Cartan algebra (in case) representing generator $${\mathbb {Z}}_2$$ -group symmetry Dynkin diagram. properties duality identities studied. also present flux integrals over two-dimensional submanifold. Dilatonic black hole analogs obtained Melvin-type solutions, e.g. “phantom” ones, considered. phantom holes described fluxbrane under consideration.