  We study the neutron star axisymmetric critical solution previously found in
the numerical studies of neutron star mergers. Using neutron star-like initial
data and performing similar merger simulations, we demonstrate that the
solution is indeed a semi-attractor on the threshold plane separating the basin
of a neutron star and the basin of a black hole in the solution space of the
Einstein equations. In order to explore the extent of the attraction basin of
the neutron star semiattractor, we construct initial data phase spaces for
these neutron star-like initial data. From these phase spaces, we also observe
several interesting dynamical scenarios where the merged object is supported
from prompt collapse. The properties of the critical index of the solution, in
particular, its dependence on conserved quantities, are then studied. From the
study, it is found that a family of neutron star semi-attractors exist that can
be classified by both their rest masses and ADM masses.
