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A comprehensive analysis of guided modes of a novel type of a planar Bragg reflection waveguide which consists of a low refractive index guiding layer sandwiched between two finite aperiodic mirrors is presented. The layers in the mirrors are aperiodically arranged according to the Kolakoski substitution rule. In such a waveguide light is confined inside the core by Bragg reflection, while dispersion characteristics of guided modes strongly depend on aperiodicity of the cladding. Using the transfer matrix formalism bandgap conditions, dispersion characteristics and mode profiles of the guided modes of such a waveguide are studied on the GaAs/AlAs and Si/SiO2 epitaxial platforms, which are compatible with hybrid and heteroepitaxial frameworks of silicon photonics.