Minimal Models RG flows:non-invertible symmetries & non-perturbative description

In this letter we continue the investigation of RG flows between minimal models that are protected by non-invertible symmetries. RG flows leaving unbroken a subcategory of non-invertible symmetries are associated with anomaly-matching conditions that we employ systematically to map the space of flows between Virasoro Minimal models beyond the $\mathbb{Z}_2$-symmetric proposed recently in the literature. We introduce a family of non-linear integral equations that appear to encode the exact finite-size, ground-state energies of these flows, including non-integrable cases, such as the recently proposed $\mathcal{M}(k q + I,q) \to \mathcal{M}(k q - I,q)$. Our family of NLIEs encompasses and generalises the integrable flows known in the literature: $\phi_{(1,3)}$, $\phi_{(1,5)}$, $\phi_{(1,2)}$ and $\phi_{(2,1)}$. This work uncovers a new interplay between exact solvability and non-invertible symmetries. Furthermore, our non-perturbative description provides a non-trivial test for all the flows conjectured by anomaly matching conditions, but so far not-observed by other means.