Integrability and dark states of the XX spin-1 central spin model in a transverse field

It was recently shown that, for central spin-1/2 and central spin-1, the XX central spin model is integrable in the presence of a magnetic field oriented perpendicular to the XY plane in which the coupling exists. In the spin-1/2 case, it was also shown, through an appropriate limit of the non-skew symmetric XXZ Richardson-Gaudin models, that it remained integrable even when the magnetic field is tilted to contain an in-plane component. Although the model has not yet been shown to explicitly belong to a known class of Richardson-Gaudin models, we show, in this work, that the spin-1 case also remains integrable in a tilted magnetic field. We do so by writing explicitly the complete set of conserved charges, then showing that these operators obey polynomial relations. It is finally demonstrated numerically that dark states, for which the central spin is completely unentangled with the bath, can emerge at strong enough coupling just as they do in the central spin-1/2 model in an arbitrarily oriented magnetic field.