Dripping from a faucet is studied, where by cutting diagonally the tip breaks the cylindrical symmetry. We measured long dripping time series recorded with a laser-beam apparatus and continuous change in drop position using a high-speed camera. It is found that the added degree of freedom produces a transversal oscillation of a pending drop, which couples with a vertical oscillation induced by the breaking off of the previous drop. As a consequence dripping times shorten and dripping patterns regularize. The effect is attributed to the "reduced" contact circle and to decreased frequency of the vertical oscillations of the residue. A very complex flow circulation pattern within the forming drop is observed. It is suggested that geometrical shape vibrations of the pending drop take place by the development of eddies of different amount, more effective in the vicinity of the highest edge of the tip, where they dissipate more slowly. This asymmetrical liquid flow is brought about by the slanting form of the orifice and overlaps with the axial oscillations.