The analytical regularisation technique for rigorous solution of dual series equations in diffraction theory for structures, which consist of coaxial perfect (perfectly conducting, rigid or soft) conical sections, is proposed. The conical sections do not form biconical regions and the continuation of their generatrices has a single crossing point. The proposed approach is based on the mode matching technique as well as on establishing of the rule for correct transition to the infinite systems of linear algebraic equations and on receiving the solutions, which provide the fulfilment of all the necessary conditions for all the boundary value problems considered here: Neumann, Dirichlet, scalar electromagnetic problem. These systems are proved to be regularized by a pair of operators, which consist of the convolution type operator and the corresponding inverse one. The elements of the inverted operator are founded analytically using the factorization procedure for kernel functions for each boundary value problem. The peculiarities of the far field formation by one- and two-section cones in the case of their axially symmetric excitation by TM- electromagnetic wave are analyzed numerically.