The S-type asymptotic giant branch (AGB) star $pi^{1}$ Gruis has a known companion at a separation of $approx$400 AU. The envelope structure, including an equatorial torus and a fast bipolar outflow, is rarely seen in the AGB phase and is particularly unexpected in such a wide binary system. Therefore a second, closer companion has been suggested, but the evidence is not conclusive. The new ALMA $^{12}$CO and $^{13}$CO $J$=3-2 data, together with previously published $^{12}$CO $J$=2-1 data from the Submillimeter Array (SMA), and the $^{12}$CO $J$=5-4 and $J$=9-8 lines observed with Herschel/Heterodyne Instrument for the Far-Infrared (HIFI), is modeled with the 3D non-LTE radiative transfer code SHAPEMOL. The data analysis clearly confirms the torus-bipolar structure. The 3D model of the CSE that satisfactorily reproduces the data consists of three kinematic components: a radially expanding torus with velocity slowly increasing from 8 to 13 km s$^{-1}$ along the equator plane; a radially expanding component at the center with a constant velocity of 14 km s$^{-1}$; and a fast, bipolar outflow with velocity proportionally increasing from 14 km s$^{-1}$ at the base up to 100 km s$^{-1}$ at the tip, following a linear radial dependence. The results are used to estimate an average mass-loss rate during the creation of the torus of 7.7$times$10$^{-7}$ M$_{odot}$ yr$^{-1}$. The total mass and linear momentum of the fast outflow are estimated at 7.3$times$10$^{-4}$ M$_{odot}$ and 9.6$times$10$^{37}$ g cm s$^{-1}$, respectively. The momentum of the outflow is in excess (by a factor of about 20) of what could be generated by radiation pressure alone, in agreement with recent findings for more evolved sources. The best-fit model also suggests a $^{12}$CO/$^{13}$CO abundance ratio of 50. Possible shaping scenarios for the gas envelope are discussed
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