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Photonic bound states in the continuum (BICs) have recently been studied in various systems and have found wide applications in sensors, lasers, and filters. Applying BICs in photonic integrated circuits enables low-loss light guidance and routing in low-refractive-index waveguides on high-refractive-index substrates, which opens a new avenue for integrated photonics with functional single-crystal materials. Here, we demonstrate high-quality integrated lithium niobate microcavities inside which the photonic BIC modes circulate and further modulate these BIC modes acousto-optically by using piezoelectrically actuated surface acoustic waves at microwave frequencies. With a high acousto-optic modulation frequency, the acousto-optic coupling is well situated in the resolved-sideband regime. This leads to coherent coupling between microwave and optical photons, which is exhibited by the observed electro …