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We present the first detailed kinematical analysis of the planetary nebula Abell 63, which is known to contain the eclipsing close-binary nucleus UU Sge. Abell 63 provides an important test case in investigating the role of close-binary central stars on the evolution of planetary nebulae. Longslit observations were obtained using the Manchester echelle spectrometer combined with the 2.1-m San Pedro Martir Telescope. The spectra reveal that the central bright rim of Abell 63 has a tube-like structure. A deep image shows collimated lobes extending from the nebula, which are shown to be high-velocity outflows. The kinematic ages of the nebular rim and the extended lobes are calculated to be 8400 +/- 500 and 12900 +/- 2800 yr, respectively, which suggests that the lobes were formed at an earlier stage than the nebular rim. This is consistent with expectations that disc-generated jets form immediately after the common envelope phase. A morphological-kinematical model of the central nebula is presented and the best-fitting model is found to have the same inclination as the orbital plane of the central binary system; this is the first proof that a close-binary system directly affects the shaping of its nebula. A Hubble-type flow is well-established in the morphological-kinematical modelling of the observed line profiles and imagery. Two possible formation models for the elongated lobes of Abell 63 are considered, (i) a low-density, pressure-driven jet excavates a cavity in the remnant asymptotic giant branch (AGB) envelope; (ii) high-density bullets form the lobes in a single ballistic ejection event.