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Amplitude models are applied to studies of resonance structure in $D^0\toK^0_S K^- \pi^+$ and $D^0\to K^0_S K^+ \pi^-$ decays using $pp$ collision datacorresponding to an integrated luminosity of $3.0\,\mathrm{fb}^{-1}$ collectedby the LHCb experiment. Relative magnitude and phase information is determined,and coherence factors and related observables are computed for both the wholephase space and a restricted region of $100\,\mathrm{MeV/}c^2$ around the$K^{*}(892)^{\pm}$ resonance. Two formulations for the $K\pi$ $S$-wave areused, both of which give a good description of the data. The ratio of branchingfractions $\mathcal{B}(D^0\to K^0_S K^+ \pi^-)/\mathcal{B}(D^0\to K^0_S K^-\pi^+)$ is measured to be$0.655\pm0.004\,(\textrm{stat})\pm0.006\,(\textrm{syst})$ over the full phasespace and $0.370\pm0.003\, (\textrm{stat})\pm0.012\,(\textrm{syst})$ in therestricted region. A search for $CP$ violation is performed using the amplitudemodels and no significant effect is found. Predictions from $SU(3)$ flavorsymmetry for $K^{*}(892)K$ amplitudes of different charges are compared withthe amplitude model results.

Studies of the resonance structure in $D^0\to K^0_S K^{\pm}\pi^{\mp}$ decays