We explain how integrable models can be connected to both $\mathcal{N}=2$ supersymmetric gauge theory and black hole perturbation theory. This important fact relies on the analysis of the connection coefficients of the differential equations which all this theories happen to share. In particular we prove that quasinormal modes of black holes are given by quantisation conditions on gauge periods or, equivalently, integrability $Q$ functions. Besides, this allows us to find a new efficient method to characterise both analytically and numerically the quasinormal modes: analytically, we shed light on the gauge theory application; numerically, we compare it with other methods. For simplicity and limits of space we restrict the discussion to the simplest case of Liouville integrable model/pure $SU(2)$ gauge theory/D3 brane gravitation background triad.