  The damped oscillating structures (OS) were recently revealed in the proton
"effective" form factor (FF) data. For the time being they can be neither
confirmed nor disproved by investigations of timelike data on the individual
proton electric and proton magnetic FFs because their precision and reliability
(especially of the proton electric FF data) has not achieved required level for
this aim. On the other hand, conjectures that the OS are direct manifestations
of the quark-gluon structure of the proton indicate that they must not be
specific only for the proton and neutron, but that they should be present also
for other hadrons. This opens a plausibility to find damped oscillatory
structures also from the EM FFs data of such hadrons, for which adequate EM FFs
data exist, by using the same procedure as for the proton. Consequently in this
paper damped oscillatory structures are investigated in the EM FFs data of the
charged and neutral $K$-mesons to be extracted from the corresponding
production cross sections, $\sigma^{bare}_{tot}(e^+e^-\to K^+ K^-)$ measured
from the threshold up to 64 GeV$^2$ and $\sigma^{bare}_{tot}(e^+e^-\to K_s
K_L)$ measured from the threshold up to 9.5 GeV$^2$ of the total c.m. energy
squared. The following results have been obtained. If the charged and neutral
K-meson EM FFs timelike data are described by the three parametric formula by
means of which OS have been revealed from the "effective" proton FF data then
OS appear. If physically well founded Unitary and Analytic model of the K-meson
EM structure is used for a description of the charged K-meson EM FFs data, no
OS are visible. However, in the case of the neutral K-meson EM FF data one
cannot make a definite decision. The overall results indicate that OS obtained
from the "effective" proton FF data are likely an artefact of the three
parametric formula which does not describe these data well.
