Generalized composite fluxbrane solutions for a wide class of intersection rules are obtained. The solutions are defined on a manifold which contains a product of n Ricci-flat spaces M1 × ⋯ × M n with one-dimensional M1. They are defined up to a set of functions Hs obeying nonlinear differential equations equivalent to Toda-type equations with certain boundary conditions imposed. A conjecture on polynomial structure of governing functions Hs for intersections related to semisimple Lie algebras is suggested. This conjecture is valid for Lie algebras: Am, Cm+1, m ≥ 1. For simple Lie algebras the powers of polynomials coincide with the components of the dual Weyl vector in the basis of simple roots. Explicit formulae for A1 ⊕ ⋯ ⊕ A1 (orthogonal), 'block-orthogonal' and A 2 solutions are obtained. Certain examples of solutions in D = 11 and D = 10 (II A) supergravities (e.g. with A2 intersection rules) and Kaluza-Klein dyonic A2 flux tube, are considered.