Using complex variable methods and conformal mapping techniques, we establish the existence of coated cavities of various shapes that do not disturb the warping function in a host elliptical bar. These cavities are known as `partly neutral cavities'. Our results show that the two axes corresponding to the neutral elliptical coating are parallel to those of the host elliptical bar and that the centre of the elliptical coating can be located arbitrarily on the major axis of the host bar. Examples of neutral coated non-elliptical cavities are provided.