  In conical refraction (CR), a focused Gaussian input beam passing through a
biaxial crystal and parallel to one of the optic axes is transformed into a
pair of concentric bright rings split by a dark (Poggendorff) ring at the focal
plane. Here, we show the generation of a CR transverse pattern that does not
present the Poggendorff fine splitting at the focal plane, i.e. it forms a
single light ring. This light ring is generated from a non-homogeneously
polarized input light beam obtained by using a spatially inhomogeneous
polarizer that mimics the characteristic CR polarization distribution. This
polarizer allows modulating the relative intensity between the two CR light
cones in accordance with the recently proposed dual--cone model of the CR
phenomenon. We show that the absence of interfering rings at the focal plane is
caused by the selection of one of the two CR cones.
