  In this paper a neutron star with an inner core which undergoes a phase
transition, which is characterized by conformal degrees of freedom on the phase
boundary, is considered. Typical cases of such a phase transition are e.g.
quantum Hall effect, superconductivity and superfluidity. Assuming the
mechanical stability of this system the effects induced by the conformal
degrees of freedom on the phase boundary will be analyzed. We will see that the
inclusion of conformal degrees of freedom is not always consistent with the
staticity of the phase boundary. Indeed also in the case of mechanical
equilibrium there may be the tendency of one phase to swallow the other. Such a
shift of the phase boundary would not imply any compression or decompression of
the core. By solving the Israel junction conditions for the conformal matter,
we have found the range of physical parameters which can guarantee a stable
equilibrium of the phase boundary of the neutron star. The relevant parameters
turn out to be not only the density difference but also the difference of the
slope of the density profiles of the two phases. The values of the parameters
which guarantee the stability turn out to be in a phenomenologically reasonable
range. For the parameter values where the the phase boundary tends to move, a
possible astrophysical consequence related to sudden small changes of the
moment of inertia of the star is briefly discussed.
