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At the present time, for the two-dimensional oscillation case well studied are the natural oscillation spectra waveguide branchings as the half-open systems. In the branchings formed both with rectangular below-cutoff waveguides and with circular and radial waveguides, such oscillations of branching are divided into two classes: free oscillations of branching (FOB) and oscillations that are analogous to the ones in the waveguide-dielectric resonators (WDR). For FOB both waveguides forming the branching, and the waveguide crossing region are below-cutoff for all types of waves. Below-cutoff dimensions of the waveguides for the resonating types of waves and possibility of existence of propagating waves in the waveguide crossing region are the necessary conditions for the WDR existence.