Abstract
We prove that under some conditions every nonparametric capillary surface which has a central fan (of radial limits at a point O) can be perturbed with respect to the contact angle and the perturbed surfaces continue to have central fans. In particular, any nonparametric capillary surface which is symmetric with respect to a vertical plane through O and has a central fan may be perturbed (with respect to the contact angle) in a nonsymmetric manner and the resulting capillary surfaces will not be symmetric with respect to the vertical plane but will continue to have central fans.