  Neutron stars are the densest objects in the Universe. In this paper we
consider so-called inner crust - the layer, where neutron-excess nuclei are
immersed into degenerate gas of electrons and sea of quasi-free neutrons. It
was generally believed that spherical nuclei become unstable with respect to
quadrupole deformations at high densities and here we consider this
instability. Within perturbative approach we show that spherical nuclei with
equilibrium number density are, in fact, stable with respect to infinitesimal
quadrupole deformation. This is due to background of degenerate electrons and
associated electrostatic potential which maintain stability of spherical
nuclei. However, if the number of atomic nuclei per unit volume is much less
than the equilibrium value, instability can arise. To avoid confusion we stress
that our results are limited to infinitesimal deformations and do not guaranty
strict thermodynamic stability of spherical nuclei. In particular, they does
not exclude that substantially non-spherical nuclei (so-called pasta phase)
represent thermodynamic equilibrium state of the densest layers of neutron star
crust. Rather our results points that spherical nuclei can be metastable even
if they are not energetically favourable and the timescale of transformation of
spherical nuclei to the pasta phases should be estimated subsequently.
