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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., Comment: 10 pages, Latex, 1 figure; 5th version: the abstract in the Latex file is corrected. arXiv admin note: text overlap with arXiv:1706.07856

This topical review deals with a multidimensional gravitational model containing dilatonic scalar fields and antisymmetric forms. The manifold is chosen in the form M = M_0 x M_1 x ...x M_n, where M_i are Einstein spaces (i >0). The sigma-model approach and exact solutions in the model are reviewed and the solutions with p-branes (e.g. Majumdar-Papapetrou-type, cosmological, spherically symmetric, black-brane and Freund-Rubin-type ones) are considered., 63 pages, Latex, topical review, few typos are eliminated

The block-orthogonal generalization of the Majumdar-Papapetrou type solutions for the sigma-model studied earlier are obtained and corresponding solutions with p-branes are considered. The existence of solutions and the number of independent harmonic functions is defined by the matrix of scalar products of vectors $U^s$, governing the sigma-model target space metric. For orthogonal $U^s$, when target space is a symmetric homogeneous space, the solutions reduce to the previous ones. Two special classes of solutions with $U^s$ related to finite dimensional Lie algebras and hyperbolic (Kac-Moody) algebras are singled out and investigated. The affine Cartan matrices do not arise in the scheme under consideration. Some examples of obtained solutions and intersection rules for D=11 supergravity, related D=12 theory and extending them $B_D$-models are considered. For special multicenter solutions criterions for the existence of horizon and curvature singularity are found., Comment: 22 pages, Latex. Resubmit. to Class. and Quantum Grav. Revised version

The vacuum cosmological model on the manifold $R \times M_1 \times \ldots \times M_n$ describing the evolution of $n$ Einstein spaces of non-zero curvatures is considered. For $n = 2$ the Einstein equations are reduced to the Abel (ordinary differential) equation and solved, when $(N_1 = $dim $ M_1, N_2 = $ dim$ M_2) = (6,3), (5,5), (8,2)$. The Kasner-like behaviour of the solutions near the singularity $t_s \to +0$ is considered ($t_s$ is synchronous time). The exceptional ("Milne-type") solutions are obtained for arbitrary $n$. For $n=2$ these solutions are attractors for other ones, when $t_s \to + \infty$. For dim $ M = 10, 11$ and $3 \leq n \leq 5$ certain two-parametric families of solutions are obtained from $n=2$ ones using "curvature-splitting" trick. In the case $n=2$, $(N_1, N_2)= (6,3)$ a family of non-singular solutions with the topology $R^7 \times M_2$ is found., Comment: 21 pages, LaTex. 5 figures are available upon request (hard copy). Submitted to Classical and Quantum Gravity

We prove that Gibbons-Maeda reduction of two-component cosmology to the Toda lattice can not be generalyzed for n-component case.

Dilatonic black hole dyon solutions with an arbitrary dilatonic coupling constant and canonical sign for the scalar field kynetic term are considered. These solutions are defined up to solutions of two master equations for moduli functions. For the solutions are extended to = ±1, where corresponds to a ghost (phantom) scalar field. Some physical parameters of the solutions: gravitational mass, scalar charge, Hawking temperature, black hole area entropy and parametrized post-Newtonian (PPN) parameters β and γ are obtained. It is shown that PPN parameters do not depend on scalar field coupling λ and . Two groups of bounds on gravitational mass and scalar charge (for a fixed and arbitrary extremality parameter ) are found by using a certain conjecture on parameters of solutions when . These bounds are verified numerically for certain examples. Using the product we are led to the well-known lower bound on mass which was obtained earlier by Gibbons, Kastor, London, Townsend and Traschen by using spinor techniques.