  In Kaluza-Klein models, we investigate soliton solutions of Einstein
equation. We obtain the formulas for perihelion shift, deflection of light,
time delay of radar echoes and PPN parameters. We find that the solitonic
parameter k should be very big: |k|\geq 2.3\times10^4. We define a soliton
solution which corresponds to a point-like mass source. In this case the
soliton parameter k=2, which is clearly contrary to this restriction. Similar
problem with the observations takes place for static spherically symmetric
perfect fluid with the dust-like equation of state in all dimensions. The
common for both of these models is the same equations of state in our three
dimensions and in the extra dimensions. All dimensions are treated at equal
footing. To be in agreement with observations, it is necessary to break the
symmetry between the external/our and internal spaces. It takes place for black
strings which are particular examples of solitons with k\to \infty. For such k,
black strings are in concordance with the observations. Moreover, we show that
they are the only solitons which are at the same level of agreement with the
observations as in general relativity. Black strings can be treated as perfect
fluid with dust-like equation of state p_0=0 in the external/our space and very
specific equation of state p_1=-(1/2)\epsilon in the internal space. The latter
equation is due to negative tension in the extra dimension. We also demonstrate
that dimension 3 for the external space is a special one. Only in this case we
get the latter equation of state. We show that the black string equations of
state satisfy the necessary condition of the internal space stabilization.
Therefore, black strings are good candidates for a viable model of
astrophysical objects (e.g., Sun) if we can provide a satisfactory explanation
of negative tension for particles constituting these objects.
