  A numerical rotating neutron star solver is used to study the temporal
evolution of accreting neutron stars using a multi-polytrope model for the
nuclear equation of state named ACB5. The solver is based on a quadrupole
expansion of the metric, but confirms the results of previous works, revealing
the possibility of an abrupt transition of a neutron star from a purely
hadronic branch to a third-family branch of stable hybrid stars, passing
through an unstable intermediate branch. The accretion is described through a
sequence of stationary rotating {stellar} configurations which lose angular
momentum through magnetic dipole emission while, at the same time, gaining
angular momentum through mass accretion. The model has several free parameters
which are inferred from observations. The mass accretion scenario is studied in
dependence on the effectiveness of angular momentum transfer which determines
at which spin frequency the neutron star will become unstable against
gravitational collapse to the corresponding hybrid star on the stable
third-family branch. It is conceivable that the neutrino burst which
accompanies the deconfinement transition may trigger a pulsar kick which
results in the eccentric orbit. A consequence of the present model is the
prediction of a correlation between the spin frequency of the millisecond
pulsar in the eccentric orbit and its mass at birth.
