  We identify combinations of observables for rotating neutron stars that can
one day bear on the question of whether there can be first order phase
transitions in the neutron matter therein. We employ the Hartle-Thorne theory
for stationary, rotating neutron stars at conventional angular velocities (in
the pulsar and millisecond pulsar ranges) and extract three-dimensional
sections of the ellipticity or the dynamical angular momentum as function of
the star's mass and angular velocity. An eventual first order phase transition
in the equation of state (EoS) leaves a clear ridge (nonanalyticity) in these
observables, akin to the sudden kink in popular mass-radius diagrams for static
stars. Finally, we observe that static neutron stars in General Relativity (GR)
will fail to be compact enough for the light ring's position at r=3M to be
outside the star, except for the most extreme equations of state. The outer
light ring of a rotating star might however be formed unless the EoS softens
too much, and its eventual detection can then be used to constrain the EoS (or
the gravity theory).
