  Central spin models provide an idealized description of interactions between
a central degree of freedom and a mesoscopic environment of surrounding spins.
We show that the family of models with a spin-1 at the center and XX
interactions of arbitrary strength with surrounding spins is integrable.
Specifically, we derive an extensive set of conserved quantities and obtain the
exact eigenstates using the Bethe ansatz. As in the homogenous limit, the
states divide into two exponentially large classes: bright states, in which the
spin-1 is entangled with its surroundings, and dark states, in which it is not.
On resonance, the bright states further break up into two classes depending on
their weight on states with central spin polarization zero. These classes are
probed in quench dynamics wherein they prevent the central spin from reaching
thermal equilibrium. In the single spin-flip sector we explicitly construct the
bright states and show that the central spin exhibits oscillatory dynamics as a
consequence of the semilocalization of these eigenstates. We relate the
integrability to the closely related class of integrable Richardson-Gaudin
models, and conjecture that the spin-$s$ central spin XX model is integrable
for any $s$.

Strongly Correlated Electrons;Mesoscale and Nanoscale Physics;Statistical Mechanics