  The main purpose of our paper is to construct a viable Kaluza-Klein model
satisfying the observable constraints. To this end, we investigate the
six-dimensional model with spherical compactification of the internal space.
Background matter is considered in the form of a perfect fluid with non-linear
equations of state both in the external/our and internal spaces and the model
is set to include an additional bare cosmological constant $\Lambda_6$. In the
weak-field approximation, the background is perturbed by pressureless
gravitating mass that is a static point-like particle. The non-linearity of the
equations of state of a perfect fluid makes it possible to solve simultaneously
a number of problems. The demand that the parameterized post-Newtonian
parameter $\gamma$ be equal to 1 in this configuration, first, ensures
compatibility with gravitational tests in the Solar system (deflection of light
and time delay of radar echoes) at the same level of accuracy as General
Relativity. Second, it translates into the absence of internal space variations
so that the gravitational potential coincides exactly with the Newtonian one,
securing the absence of the fifth force. Third, the gravitating mass remains
pressurless in the external space as in the standard approach to
non-relativistic astrophysical objects and meanwhile, acquires effective
tension in the internal space.
