n = 0 and (5). The statement (5) about the linear nature of Zeno-states is far from trivial. Indeed the expression for Z , which for the system with binary potential interaction Φ( r$_{12}$ ) reads as (see e.g. [2]): where g$_{2}$ is the pair correlation function. Therefore the linearity of the Zeno states is due to quite specific structure of the dependence of correlation function g$_{2}$ ( r ; T, n ) on the thermodynamic state. Below, following the arguments of previous work [15], we show that these facts can be casted into elegant geometrical formulation and expand them into general case of the short range power-like attractive potentials in d dimensions. We show that the locus of the CP can be estimated using the correspondence between the scaling properties of the Hamiltonians for the lattice gas and the fluids with the power-like interactive potentials. As was shown in [15] the linearities of (2) and (5) can be derived on the basis of the assumption of the existence of the global isomorphism of the real liquid-gas part of the phase diagram of the lattice gas model. Note that the line x = 1 of the LG can be thought of as the analog of the Zeno-line. Indeed the pair correlation function g$_{2}$ ( r ) of the LG vanishes identically for such “holeless“ states according to the definition. The same is true for the state with empty sites x = 0. Thus the line x = 0 can be identified with the zeroth density axes n = 0 of the real fluid, while the line x = 1, obviously, can be identified with the Zeno-line Z = 1. To conserve the linearity and the adjacency properties of the characteristic elements the isomorphism between LG and the fluid should be chosen in the class of the projective mappings [16] and has the form: where The coordinates of the CP for the liquid are: Z = P n T = 1 − 2 π n 3 T ∫ r $_{3}$∂ Φ( r ) ∂ r g$_{2}$ ( r ; n, T ) d r , (7) n = n$_{b}$ x 1 + z t , T = T$_{Z}$ z t 1 + z t , (8) z = T$_{c}$ T$_{Z}$ − T$_{c}$ . (9) n$_{c}$ = n$_{b}$ 2 ( 1 + z ) , T$_{c}$ = T$_{Z}$ z 1 + z , (10) 4 THE ZENO-LINE AND THE GLOBAL ISOMORPHISM