We introduce a new type of models for two-component systems in one dimension subject to exact solutions by Bethe ansatz, where the interspecies interactions are tunable via Feshbach resonant interactions. The applicability of Bethe ansatz is obtained by fine-tuning the resonant energies, and the resulting systems can be described by introducing intraspecies repulsive and interspecies attractive couplings $c_1$ and $c_2$. This kind of systems admits two types of interesting solutions: In the regime with $c_1>c_2$, the ground state is a Fermi sea of two-strings, where the Fermi momentum $Q$ is constrained to be smaller than a certain value $Q^*$, and it provides an ideal scenario to realize BCS-BEC crossover (from weakly attractive atoms to weakly repulsive molecules) in one dimension; In the opposite regime with $c_1<c_2$, the ground state is a single bright soliton even for fermionic atoms, which reveals itself as an embedded string solution.
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