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They vastly generalize the previously proposed ones $k=I=1$ by Zamolodchikov, $k=1, I>1$ by Ahn and L\\\"assig, and $k=2$ by Dorey et al. All the other $\\mathbb{Z}_2$ preserving renormalization group flows sporadically known in the literature (e.g. $\\mathcal{M}(10,3) \\to \\mathcal{M}(8,3)$ studied by Klebanov et al) fall into our proposal (e.g. $k=3, I=1$). 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