Integral field spectroscopy of the inner region of the Galactic Center, over a field of roughly 40''*40'' was obtained at 2.06 micron (HeI) and 2.16 micron (Br gamma ) using BEAR, an imaging Fourier Transform Spectrometer, at spectral resolutions respectively of 52.9 km s-1 and 21.3 km s-1, and a spatial resolution of 0.5''. The analysis of the data was focused on the kinematics of the gas flows, traditionally called the ``Minispiral'', concentrated in the neighborhood of the central black hole, Sgr A*. From the decomposition into several velocity components (up to four) of the line profile extracted at each point of the field, velocity features were identified. Nine distinguishable structures are described: the standard Northern Arm, Eastern Arm, Bar, Western Arc, and five additional, coherently-moving patches of gas. From this analysis, the Northern Arm appears not limited, as usually thought, to the bright, narrow North-South lane seen on intensity images, but it consists instead of a weak, continuous, triangular-shaped surface, drawn out into a narrow stream in the vicinity of Sgr A* where it shows a strong velocity gradient, and a bright western rim. The Eastern Arm is split into three components (a Ribbon and a Tip, separated by a cavity, and an elongated feature parallel to the Ribbon: the Eastern Bridge). We also report extinction of some interstellar structures by other components, providing information on their relative position along the line of sight. A system of Keplerian orbits can be fitted to most of the Northern Arm, and the bright rim of this feature can be interpreted in terms of line-of-sight orbit crowding caused by the warping of the flowing surface at the western edge facing Sgr A*. These results lead to a new picture of the gas structures in Sgr A West, in which large-scale gas flows and isolated gas patches coexist in the gravitational field of the central Black Hole. The question of the origin of the ionized gas is addressed and a discussion of the lifetime of these features is presented.