Polarization effects in liquid crystal-core optical fiber waveguides have been investigated based on extremely low-birefringence liquid crystal mixture acting as an optical fiber core characterized by an index ellipsoid. In certain conditions the liquid crystal elliptical-core fiber acts as a single-polarization fiber in which only one polarization mode is guided. The paper presents first experimental results of the liquid crystal fiber in the polarimetric configuration subjected to the influence of hydrostatic pressure and electric field.