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A search for $\textit{CP}$ violation in $D^{0} \rightarrow K_{S}^{0} K^{+}\pi^{-} $ and $D^{0} \rightarrow K_{S}^{0} K^{-} \pi^{+} $ decays is reported.The search is performed using an unbinned model-independent method known as theenergy test that probes local $\textit{CP}$ violation in the phase space of thedecays. The data analysed correspond to an integrated luminosity of$5.4~$fb$^{-1}$ collected in proton-proton collisions by the LHCb experiment ata centre-of-mass energy of $\sqrt{s}=13$~TeV, amounting to approximately 950000and 620000 signal candidates for the $D^{0} \rightarrow K_{S}^{0} K^{-} \pi^{+}$ and $D^{0} \rightarrow K_{S}^{0} K^{+} \pi^{-} $ modes, respectively. Themethod is validated using $D^{0} \rightarrow K^{-} \pi^{+} \pi^{-} \pi^{+} $and $D^{0} \rightarrow K_{S}^{0} \pi^{+} \pi^{-} $ decays, where$\textit{CP}$-violating effects are expected to be negligible, and usingbackground-enhanced regions of the signal decays. The results are consistentwith $\textit{CP}$ symmetry in both the $D^{0} \rightarrow K_{S}^{0} K^{-}\pi^{+} $ and the $D^{0} \rightarrow K_{S}^{0} K^{+} \pi^{-} $ decays, with$p$-values for the hypothesis of no $\textit{CP}$ violation of 70% and 66%,respectively.

Search for $\textit{CP}$ violation in the phase space of $D^{0} \rightarrow K_{S}^{0} K^{\pm} \pi^{\mp} $ decays with the energy test