  In this paper, we consider a system of gravitating bodies in Kaluza-Klein
models with toroidal compactification of extra dimensions. To simulate the
astrophysical objects (e.g., our Sun and pulsars) with energy density much
greater than pressure, we suppose that these bodies are pressureless in the
external/our space. At the same time, they may have nonzero parameters
\omega_{(\bar\alpha -3)} \, (\bar\alpha =4,\ldots,D) of the equations of state
in the extra dimensions. We construct the Lagrange function of this many-body
system for any value of \Sigma =\sum_{\bar\alpha} \omega_{(\bar\alpha -3)}.
Moreover, the gravitational tests (PPN parameters, perihelion and periastron
advances) require negligible deviation from the latent soliton value \Sigma
=-(D-3)/2. However, the presence of pressure/tension in the internal space
results necessarily in the smearing of the gravitating masses over the internal
space and in the absence of the KK modes. This looks very unnatural from the
point of view of quantum physics.
