  The back-bending phenomenon for compact stars is studied by means of
analytical equations of state, for both constant-pressure phase transitions and
the transitions through the mixed-phase region. We restrict ourselves to the
case of normal rotating configurations, with baryon mass below the maximum
allowable baryon mass for non-rotating stars. We use high-precision 2-D
multi-domain spectral code LORENE to search the parameter space for possible
instability regions, and possible changes in the stability character of
rotating stars with phase transitions in their cores. Conditions on the density
jump in constant-pressure phase transitions, leading to the existence of the
unstable segments in the evolutionary sequences of spinning down isolated
normal neutron stars, are derived. Conjectures concerning the existence of two
disjoint families of non-rotating and rotating stationary configurations of
neutron stars are formulated. Particular case of EOSs leading to marginal
instability of static and rotating configurations is also studied: marginal
instability point in non-rotating configurations continues to exist in all
evolutionary spin-down tracks. The fate of rotating stars entering the region
of instability is discussed. The change in radius, energy release, and spin-up
associated with the corequake in rotating neutron star, triggered by the
instability, are calculated. The energy release is found to be very weakly
dependent on the angular momentum of collapsing star.
