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A novel type of aperiodic porous silicon-based multilayered structures which can be used for the electromagnetic radiation manipulation in the visible and infrared parts of spectrum are proposed in this paper. The one-dimensional aperiodic structures under consideration are arranged by stacking together two anisotropic layers according to the Kolakoski self-generation scheme. The spectral properties of the structures are theoretically investigated by using the electromagnetic field scalarization method in coupling with the finite-difference method. It is shown that the Kolakoski multilayers have both completely transparent states in the transmission spectra and pronounced omnidirectional reflectance which can used in integrated optics, optical fiber telecommunications, vertical cavity surface emitting lasers, laser cavities, etc. Also, it is shown numerically that the anisotropic nature of layered media significantly affects on optical properties of the structures.