I found following change of variables that can be also seen as a coordinate system $(y_1, y_2)$ described according to cartesian coordinates $(x_1, x_2)$:
$$y_1 = x_1 · (1+(x_1-1)\sin(\frac{\pi x_2}{2}))$$ $$y_2 = \frac{1}{ 1+(x_1-1)\sin(\frac{\pi x_2}{2})}$$
but do not match any of the common ones (polar, parabolic, hyperbolic, elliptic, bipolar...). I am interested in its name (if exists) in order to search for further info, uses...