Polymer-dispersed liquid crystal composites have been a focus of study for a long time for their unique electrooptical properties and manufacturing by "bottom-up" techniques at large scales. In this work, nematic liquid crystal oblate droplets with conical boundary conditions (CBC) under the action of electric field were studied by computer simulations and polarized optical microscopy. Droplets with CBC were shown to prefer the axial-bipolar structure, which combines a pair of boojums and

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... ar disclination on a surface. In contrast to droplets with degenerate planar boundary conditions (PBC), hybridization of the two structure types in droplets with CBC leads to a two minima energy profile, resulting into abrupt structure transition and bistable behavior of the system. The nature of the low energy barrier in droplets with CBC makes it highly sensible to an external stimuli, such as electric or magnetic fields, temperature, light, etc. In particular, the value of the electric field of the director reorientation in the droplets with CBC was found to be a few times smaller than the one for droplets with PBC, and the novel droplet state remained stable after switching off the voltage.