In this talk, I will introduce a new type of model for two-component systems in one dimension subject to exact solutions by Bethe ansatz. It describes the BCS-BEC crossover in one dimension and its integrability is obtained by fine-tuning the model parameters. The new model has rich many-body physics, where the Fermi momentum for the ground state distribution is constrained to be smaller than a certain value and the zero temperature phase diagram with an external field has a critical field strength for polarization. Also the low energy excitation spectra of the new model present robust features that can be related to solitons at BCS-BEC crossover in one dimension, as shown by the semiclassical analysis.