  Recent research into time-integrated observables has revealed a special class
of states which cap- ture the singular features of the generating functions of
those observables, as estimated by full counting statistics (FCS). In this work
we extend the results of [Phys. Rev. B 87 184303 (2013)] to the 1d anisotropic
XY -model and find a set of FCS critical points associated with the
time-integrated transverse magnetization and anisotropy. We show dynamical
phase transitions (DPTs) as defined in [Phys. Rev. Lett. 110 135704 (2013)] do
not emerge on quenching the states associated with the time-integrated
anisotropy across FCS critical points. We also study the timescales required to
prepare the associated special states of the transverse magnetization using
appropriate Markovian baths and find they are independent of the number of
spins in the chain. However the probability to evolve to such a state decreases
drastically with increasing system size due to dissipation. Thus when preparing
such states it is preferable to use few body systems and to reach the
thermodynamic limit it is necessary to use an ancillary system in conjunction
with the Markovian baths.
