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We introduce an exactly-solvable model to study the competition between theLarkin-Ovchinnikov-Fulde-Ferrell (LOFF) and breached-pair superfluid instrongly interacting ultracold asymmetric Fermi gases. One can thus investigatehomogeneous and inhomogeneous states on an equal footing and establish thequantum phase diagram. For certain values of the filling and the interactionstrength, the model exhibits a new stable exotic pairing phase which combinesan inhomogeneous state with an interior gap to pair-excitations. It is proventhat this phase is the exact ground state in the strong coupling limit, whilenumerical examples demonstrate that also at finite interaction strength it canhave lower energy than the breached-pair or LOFF states.Comment: Revised version accepted for publicatio

Integrable Models for Asymmetric Fermi Superfluids: Emergence of a New Exotic Pairing Phase