Central spin models provide an idealized description of interactions between a central degree of freedom and a mesoscopic environment of surrounding spins. I will present the exact solution for a family of models with XX interactions of arbitrary strength between the central spin and the surrounding spins, and show how the system supports an extensive number of conserved charges. The eigenstates divide into two exponentially large classes: bright states, in which the central spin is entangled with its surroundings, and dark states, in which it is not. This eigenstate structure and the integrability can be probed in quench dynamics, where the system does not reach thermal equilibrium, which can in turn be related to semilocalization of the eigenstates.