  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
